Constructing Ramsey Graphs from Boolean Function Representations
Parikshit Gopalan
College of Computing,
Georgia Institute of Technology,
Atlanta, GA 30332, USA.
[email protected]
May 1, 2006
Abstract
Explicit construction of Ramsey graphs or graphs with no large clique or independent set, has remained
a challenging open problem for a long time. While Erdős’ probabilistic argument shows the existence of
graphs on
vertices with no clique or independent set of size , the best explicit constructions achieve a
far weaker bound. Constructing Ramsey graphs is closely related to polynomial representations of Boolean
functions; a low degree representation for the OR function can be used to explicitly construct Ramsey graphs
[17].
We generalize the above relation by proposing a new framework. We propose a new definition of OR representations: a pair of polynomials represent the OR function if the union of their zero sets contains all points
in except the origin. We give a simple construction of a Ramsey graph using such polynomials. Furthermore, we show that all the known algebraic constructions, ones to due to Frankl-Wilson [12], Grolmusz
[17] and Alon [2] are captured by this framework; they can all be derived from various OR representations
of degree based on symmetric polynomials.
Thus the barrier to better Ramsey constructions through such algebraic methods appears to be the construction of lower degree representations. Using new algebraic techniques, we show that better bounds cannot
be obtained using symmetric polynomials.
Supported by NSF grant CCR-3606B64
1 Introduction
This paper studies a problem at the intersection of combinatorics and computational complexity.
The combinatorial problem is that of explicitly constructing Ramsey graphs. Ramsey’s theorem shows
that every graph on vertices has either a clique or an independent set of size . In his seminal 1947 paper
introducing the probabilistic method, Erdős showed that there exist graphs with vertices where [11]. He posed the question of constructing such Ramsey graphs explicitly and offered a prize of
"!! for it. This is a central open problem in explicit combinatorial constructions; the best known constructions
to date are far from the probabilistic bound. The first breakthrough on this problem was due to Frankl and Wilson
in 1981 [12]; their construction gives #
$%'& )(+*,- . For over two decades, there was no improvement
on this bound despite much effort. However there were other constructions known due to Grolmusz and Alon
[17, 2] that achieved exactly the same bound, and also extended to the problem of constructing multi-color
Ramsey graphs, which is to . -color the edges of the complete graph so that there is not large monochromatic
clique. At first sight, the construction of Grolmusz is quite different from that of Alon and Frankl-Wilson, yet it
gives exactly the same bound. All three constructions use algebraic techniques, though in different ways. Very
recently in 2006, the Frankl-Wilson bound was beaten by a new construction due to Barak, Rao, Shaltiel and
Wigderson [8] which relies on machinery from pseudorandomness.
The complexity problem is to prove tight degree bounds for polynomials computing Boolean functions over
/10
. A central open problem in circuit complexity is to show lower bounds for ACC, the class of circuits
with And, Or and Mod gates. As a first step towards this goal, Barrington, Beigel and Rudich (BBR) studied
polynomial representations of Boolean functions modulo composites [9]. They found surprisingly that such
/32
/54
representations are much more powerful over
than over
when 6 is prime. They showed that the OR
/<2
function can be represented by symmetric polynomials of degree 78:9 ; over . In contrast an => lower
/14
bound is known for the degree of such polynomials over . BBR proved a matching = 9 ; lower bound for
/?2
symmetric polynomials representing the OR function over , and asked if better representations exist using
asymmetric polynomials. Tardos and Barrington proved a lower bound of =A@CBD5 [22]. This is the best lower
bound known for any function, despite much effort [16, 23, 15, 3, 10]. The main open question in this area is
whether asymmetric polynomials can give lower degree representations of symmetric functions than symmetric
polynomials.
A surprising connection between these two problems was discovered by Grolmusz, who used the OR polynomials of BBR to construct Ramsey graphs [17, 18]. As an intermediate step, he constructed a set system of
size FEHGJIK on elements where all set sizes are ! mod 6 but all intersections are non-zero mod L , settling an
open problem in extremal set theory. He constructed Ramsey graphs from this set system and showed that lower
degree OR representations mod 6 would give better Ramsey graphs.
1.1 Our Results
Our work generalizes and extends the connection between OR polynomials and Ramsey graphs. We propose
a new definition of an OR representation: a pair of polynomials represent the OR function on variables if
the union of their zero sets contains all points in M!NO'P except the origin. We give a simple construction of a
Ramsey graph from such representations. This viewpoint based on OR polynomials unifies the constructions of
Frankl-Wilson, Alon and Grolmusz: they can all be derived from various OR representations of degree 7QR9 based on symmetric polynomials. Thus the barrier to better Ramsey constructions through algebraic techniques
appears to be the construction of lower degree representations. On one hand, since the best lower bound for any
of these representations is only =A@CBD there is the possibility of better constructions. On the other hand, we
show that further improvements cannot come from representations using symmetric polynomials; we prove an
= 9 ; lower bound for such representations.
2
1.1.1
Ramsey Graphs from OR Representations:
; denote a vector of variables and ; denote a Boolean vector. The
Let
I
I
following definition of Boolean function representation modulo was introduced by BBR [9].
/
0 weakly represents the function mod Definition 1 Polynomial Q if for M!NO'P , if
then Q Q B .
We propose the following definition of an OR representation.
/ 4 / #%
$
and !8
"
represent the OR function on variables if
Definition 2 Polynomials QA!N
:! $8B&36
and Q
! A!N:! $ B('
and for ) M!NO'P +* A!N:! Q
!, B?6
or 8
! !- B.'
where 6;/' are primes. The degree of the representation is 0(1.243 65"DF /65"D 7! .
One can combine the two polynomials using the Chinese Remainder Theorem (CRT) to get a single polynomial that weakly represents OR mod 68' . However the specific choice of values output by the weak representation
is important for our application. The construction of BBR gives a degree 78 9 OR representation using poly/ 9
$
/ :
nomials over
and . A simple representation of degree 7Q 9 with ;
8
6 '=< using polynomials over
/ 4
/ #
$
and
can be derived from Alon’s construction. This highlights another difference about our definition and
weak representations: for Ramsey constructions, we are not restricted to any fixed moduli 6 and ' , we are free
to choose them in any way, (possibly as functions of ) so that the degree is minimized as a function of .
We give a simple Ramsey construction based on OR representations: the vertex set is M!NO'P and we add
edge to if >? is in the zero set of Q , where @>? denotes the symmetric difference of and .
In order to bound #
and , we use the notion of representations of graphs over spaces of polynomials
introduced by Alon [2]. The idea is to assign polynomials to the vertices of so that the polynomials assigned
to vertices in a clique are linearly independent.
Definition 3 Let 87A /B be a graph and C be a set polynomials in variables over field D . A polynomial
representation of over D is an assignment of a polynomial FE "@C and a point GE=HD to IA where:
For each I@A , JE KE ! .
If L IN
HB then JE KMN ! .
It is easy to see that ;0ONP
QCQ which is the dimension of the D vector space spanned by polynomials
/<4
in C . We use the polynomial Q to construct a representation of over
and !Q to construct a represen/ #
$
tation of over . The Frankl-Wilson construction can also be viewed in this framework, where we represent
/ 4
and over R . However, quoting Alon ‘It seems that this construction does not extend to the case of
over
more than 2 colors’ [2]. We propose a definition of OR representation which leads to such an extension.
/ 4 S4
/ 4UT and !Q
"
V represent the OR function on variables if
Definition 4 Polynomials :! != B&36XW and 8
! A!N:! ,
!,8B&?6XY
QA!N
and for ) M!NO'P
*
A!N:! where 6 is prime and Z \[]
!, B?68W or 8
! != B&36XY
.The degree of the representation is 01V243 65"DF /65"D 7! .
3
To differentiate the representations of Definitions 2 and 4, we refer to them as prime representations and prime9
power representations respectively. The Frankl-Wilson construction can be used to show that for 6 < ,
there exist OR representations of degree 78 9 . The interesting feature of this representation is that it does not
use the Chinese Remainder Theorem (CRT). The construction of Ramsey graphs from prime-power representations stays the same; the difference is in the analysis. For this, we introduce polynomial representations of / 4 S
over
.
/
Definition 5 Let 7A /B be a graph and C a set of polynomials in variables over . A polynomial represen/54S
/
is an assignment of a polynomial E- @C and a point GE=
tation of over
to I A s.t.:
For each I@A , JE KE ! B?6 W .
If LINHB then JE KMN ! B36 W .
We show that the polynomials assigned to a clique are linearly independent over R so is bounded
by the dimension of the R -vector space spanned by C . Like polynomial representations of graphs over R ,
/ 4S
assign linearly independent polynomials over R to vertices in a clique. A crucial
representations over
/ 4 S
difference is that representations over R tensor, those over
do not. This means that if we have sets of
9
9
9
9
C
H
polynomials C
and C
that represent and over R , then C
represents over R for an
I
I
I
I
appropriate definition of graph product (see [2] for definitions and proofs). This is important for the original
application of these representations, which was to bound the Shannon capacity of the graph. However, this
property implies that one cannot get low dimensional representations of both and over R , since / 4 S
always has a large clique. But since
has zero divisors, we lose this tensor product property, so we can
/<4 S
simultaneously get low dimensional representations of and over
.
We could restate this argument from the viewpoint of OR representations. We cannot get low degree prime
/4 6 1' since then Q !Q is ! at every point in M!NO'P except the origin,
representations by taking @
and such a polynomial requires degree . But this argument does not extend to prime-power representations
because of zero-divisors.
All the OR representations above achieve a bound of 7Q 9 ; using symmetric polynomials. Plugging them
into the simple construction above gives #
% & (+*, as opposed to the best bound of % & )(+*,- .
I
However, by massaging the polynomials and working with set intersections as opposed to distances, we can get
exactly the constructions of Frankl-Wilson, Grolmusz and Alon. Here are some advantages of our unified view
of these constructions:
It places the constructions of Alon and Frankl-Wilson in the context of OR polynomials, and raises the
possibility of getting better constructions from low degree representations. The notions of prime-power
representations of graphs and Boolean functions arising from the Frankl-Wilson construction are of independent interest.
It relates the construction of Grolmusz to those of Frankl-Wilson and Alon, which look very different at
first. Our Ramsey graph construction from the OR polynomial of BBR is simple and direct. In fact it
takes some work to show that we get the same graph as Grolmusz. Viewing this construction in terms of
set intersections, we derive improved bounds for set systems with restricted intersections modulo prime
powers.
In this view, all the constructions naturally extend to multicolor Ramsey graphs. To construct . -color
/-#
/#
;
Ramsey graphs, we define OR representations involving . polynomials over
where ' /'
I
are prime powers. Taking powers of the same prime 6 extends the Frankl-Wilson construction.
4
1.1.2
Lower Bounds:
A natural question is to show tight degree bounds for OR representations. A better upper bound would lead to
better Ramsey graphs. Lower bounds are interesting from the complexity-theory viewpoint of understanding
polynomial representations over composites. For the OR function, we believe Definition 2 is the right one to
use, since it seems to eliminate dependence of the degree on the modulus 68' . Also it places the problem in the
/4
context of understanding the zero-sets of low degree polynomials over . This question has been studied in
various other contexts including low degree testing, zero-testing and derandomization (see Section 6). Primepower representations are interesting since they do not rely on the CRT. Interestingly, the =A@CBD5 lower bound
[22] also does not use the CRT, so it applies to prime-power representations too. It is possible that proving
bounds for prime-power representations is easier than the prime case.
Degree lower bounds extend a line of work in combinatorics aimed at understanding why explicit Ramsey
graphs are hard to construct, by showing limitations to various natural techniques. A conjecture of Babai states
that one cannot construct good Ramsey graphs based on the sign of a set of real polynomials. There has been
considerable progress towards proving this conjecture by Alon et al. [1, 4]. Degree lower bounds are weaker
since they say the known technique for bounding #
and does not yield good bounds, as opposed to
showing either or is large. But on the other hand, there are no good Ramsey constructions using
signs of real polynomials, while OR representations are the best technique known for this problem. Further, the
Ramsey graph constructions based on symmetric polynomials result in graphs where the vertex set is M!NO'P and
where edges are added between vertices based on the Hamming distance between them. Such graphs possess a
high degree of symmetry which is unlikely in a random graph. Our degree lower bound suggests that perhaps
such Ramsey graph constructions cannot give better parameters (see Section 6).
We show a degree = 9 lower bound for OR representations by symmetric polynomials. Thus better
representations if they exist must use asymmetric polynomials. A lower bound of =9 ; is known for symmetric
polynomials that weakly represent OR mod L [9]. Bhatnagar et al. [10] introduced the use of tools from
communication complexity for proving degree lower bounds for symmetric polynomials. One might guess that
similar arguments should work even for our definition of OR representations, but this is incorrect. In fact those
arguments will not suffice even for prime representations. The precise bound they prove, and which holds for
] N 8
6 ' . This is good enough when 6;/ ' are small, but if 6 '
8
all weak representations is 65"D) U65"DF7 !
as in Alon’s construction, this gives a bound of . One cannot hope for a stronger result since the polynomial
/#U4 #
6 ' variables over
8
of degree weakly represents OR on . Our definition restricts the values output
by the weak representation, making it possible to show bounds independent of the modulus . But exploiting
this difference calls for new techniques, beyond the periodicity based arguments used for weak representations
[9, 10].
1.1.3
Our Techniques:
While lower bounds for the prime and prime-power cases are very different, they have similar high-level structure: an algebraic part where we show that if the zero-set of the polynomial has certain structure, then the
polynomial must have high degree, and a combinatorial part where we argue that there is no good partition of
hypercube, that any partition results in one of the polynomials having high degree.
/ 4 S
For the prime-power case, we translate the problem to one about univariate polynomials modulo
. How/ 4 S
ever over
it is no longer true that a degree 0 polynomial can have only 0 roots (take W for instance); so
/ 4 S
we need new tools for degree lower bounds. Building on an algorithm for interpolation over
by the author
[14], we define a greedy sequence, which roughly is a sequence that is distributed uniformly among various
congruence classes modulo powers of 6 . We show that the longest greedy sequence in the zero-set lower-bounds
the degree of a polynomial. Then a combinatorial argument shows that in any partition of integers into and , one of them contains a long greedy sequence.
5
For the prime case, we view a symmetric polynomial acting on a 0-1 vector as a polynomial acting
on in the digits of the base 6 expansion of the weight ." following Bhatnagar et al.[10]. There it was shown
that can be used to bound 65"D) within a factor of 6 ; we introduce a notion of weighted degree of that
exactly captures the degree of . The combinatorial part of the proof uses a number theoretic lemma which
/ 4
/ #
seems of independent interest. It says that if 6;/ ' are primes, 6X' and
and
are subsets so
that every number in ; lies in mod 6 or in mod ' , then one of or has to be large.
We present our Ramsey constructions in Section 2. The lower bounds for prime-power representations are
in Section 3. In Section 5, we give an alternate construction based on set intersections that results in exactly
the constructions of Frankl-Wilson, Grolmusz and Alon. We also give improved bounds for set systems with
restricted intersections modulo prime powers. We conclude with some open problems in Section 6.
A preliminary version of this paper appears in CCC’06 [13].
1.2 Preliminaries
Let ? A!N;:! . Given M!NO'P let . denote its Hamming weight. Given M!NO'P' , >?
denotes symmetric difference, denotes the bitwise AND and 0 5 denotes Hamming distance. Let
for N;0 . Let denote the
elementary symmetric polynomial. Every
multilinear symmetric polynomial can be written as a linear combination of these polynomials. We will use
Lucas’ Theorem about binomial coefficients modulo 6 which states:
6 Fact 1 Let
!8
6 ! Then 6
6
V
B36
4 4 /
For , let the valuation of denoted I be the highest power of 6 that divides . Let I !4 . We
4 4 4 4 4 46
I I
> and I
X
I
N?I
.
have the ultrametric inequality I ];
We say a Ramsey graph 87A /B is explicit if there is a deterministic FB @ + time algorithm to compute
+ algorithm that computes the
the adjacency matrix and very explicit if there is a deterministic FB @ A@CBD
adjacency relation. We briefly describe the known explicit constructions of Ramsey graphs in chronological
order.
:
9
<
Frankl-Wilson [12]: Take 6 prime and 6 . The vertex set consists of all subsets of @ of size 6 < .
Two vertices and are adjacent if
<$
B 36 . One can bound the size of and using
well-known results from extremal set theory [12, 5].
Grolmusz [17, 18] : The main step is to construct a set system C on of size EHGJIK so that
<
! 8&
B QL
G
! 8&
,
B QL . The vertices of the graph are sets of C and are adjacent if
but
is odd. One
can bound #
and using results from extremal set theory.
:
Alon [2]: Take 6 1' to be nearly equal primes and 6 . The vertex set consists of all subsets of @ of
<
size 8
6 ' < . Two vertices and are adjacent if
8
<$
B 36 . To bound #
and , we construct
/ 4
/ #
representations of over
and over .
<
Barak [7]: Barak gives a product based construction (discovered independently by Pudlak and Rodl) where
we first explicitly search for a good Ramsey graph in a small sample space and then use the Abbot product to
! . A similar product based
get a larger graph. This gives A X
and #
& F(+*, for any
construction, but with worse parameters is given by Naor [20].
$#
0
&%'
#
(#
) * ,+ -&+
) * .+ -/+
+12304+
+1607+
1+ 5+
+1904+
80
"!
#
0
+1&:07+
+ +
<;
6
=7>
Barak-Rao-Shaltiel-Wigderson
[8]: In a recent breakthrough, Barak et al. give a construction that achieves
#
' . In fact they solve a more general problem, which is to construct bipartite Ramsey graphs.
Their construction is rather intricate and makes significant use of machinery developed for extracting randomness from weak random sources.
The first three constructions above are very explicit, the last two are merely explicit.
<
2 OR Polynomials and Ramsey graphs
In this section, we prove the correctness of the construction described in the introduction. While the graphs
obtained are not quite optimal, the construction is simple and best explains the close connections between OR
representations and Ramsey graphs.
If graph has a representation over a field D as in Definition 3, it is easy to show that 0ON7
QC / 4 S
where 0ONP
Q CQ is the dimension of the D -vector space spanned by C [2]. For representations over
, we show
that ? 0ON7
Q C where 0 N7
Q C is the dimension of the R -vector space spanned by C . The proof is by a
valuation based argument similar to one used by Babai et al. [6].
Lemma 2 If 7A5/B has a polynomial representation over
/ 4
S
, then 1;0
E P ROOF : Let be a clique. We claim that the polynomials A
R . Assume for contradiction that
E E !
for I
N7
QC .
are linearly independent over
E
/
By clearing denominators, w.m.a that EH
, and by removing common factors w.m.a that 6 does not divide
M for some L . Rearranging terms, we have
Substituting
M
,
Since M , !,8B&?6
is an edge, hence E M W
MOJM M M M
<
E E E E M
<
E E E M
E M
46 4 M ! , we have I
MOJM KM-P Z <
. But I@ and I L , then I LF
and I
! B36 W . So the RHS is divisible by 6 W , which is a contradiction. -
Construction 1 Graph 87A /B from OR polynomials.
Let AQ
F M!NO'PO .
<
<
If Q >? ! , add an edge .
Theorem 3 Given a degree
0
OR representation, graph has vertices and #
.
P ROOF : Assume that we have a prime representation. We give a polynomial representation of over
For each vertex M!NO'P , let
<
if if 7
!
/4
.
(i.e setting
Define F to be the polynomial obtained by multi-linearizing Q ;
I
I
). Note that for M !NO'P ,
1 @> Hence
F1Q - !, B?6
On the other hand, from our construction, if then Q H>
B
, !- B36 . Hence
!= B&36
/14
Thus we get a polynomial representation of over . Since the s are all multilinear polynomials
.
of degree at most 0 in variables, they lie in a vector space of dimension . This shows that Similarly, if is not an edge then Q >
,
! &
B 6 , hence !8 )>
!- B(' . Using this we
/ #
construct a representation of over
and bound #
.
/<4S
by the same argument. For prime-power representations of OR, we can represent and over
Q H
>
One can construct explicit Ramsey graphs by plugging in various OR representations described below; all of
which give 0 7Q 9 ; using symmetric polynomials. This gives a bound of % & (+*, for some constant % on
I
I
the clique size. In fact the constructions below are very explicit, since given vertices M!NO'P , the color
of the edge can be computed in time 78> .
/O
4 / #
Alon [2]: Let 6
' be primes and let 8
6 '< . Define Q "
and !Q
as
4
I
<
$
!Q
$<
#
I
(1)
." 68',< , by the CRT . !- B36 or ". ( !,8B&.' . By Fermat’s
For , since Theorem, in the former case Q !, B?6 , in the latter !Q !, B.' . Taking 6/' nearly equal gives
degree 0 :9 .
< . Define Q " /$9 and !8 " /$: V as
BBR [9]: Let !8
Q
<
<
<
<
(2)
/ 9
/ :
$
for M!NO'P , Q and !8 in fact have coefficients from
Since
and . For ;
,
."
<
. Lucas’ theorem implies that if ."V !-8B&8 then !- B8 , and if
then !Q !- B . We can choose ) s.t. 0 #' 9 for any ! [19].
." != B&
Both representations above are prime representations, we now construct prime power representations. For
ease of exposition, we restate Definition 4 of prime-power representations in terms of rational polynomials; we
omit the simple proof of equivalence.
Definition 6 Polynomials Q
! HR
QA!N
:! represent the OR function on M!NO'P if
$ B36
! A!N:! and Q
$ B36
and for ) M!NO'P +* A!N:! Q
!, B&6
or 8
! !- B?6
for a prime 6 . The degree of the representation is 01.243 65"D) / "5 DF7! .
8
>
Note that in general could be rational. When we say Q ! - B&6 , we mean is an integer
satisfying the condition. However, if and !8 !- B36 , then Q need not be an integer and vice
versa.
9
Frankl-Wilson [12]: Take 6 prime and 6
Q
<
4
. Define Q 8
! $
I
I
I
For a non-zero vector ? M!NO'P we have . $ 6
4
G K 6< hence !Q If . !, B?6 , then 6
9
<
!,
as
. If . (3)
<
=
B
B?6 . The degree is 0
<
4 I
!8
R
V
then Q
! B&6 .
=
6 < 9 9 .
We construct representations with the prime fixed and varying, analogous to [9]. Let Q !8 $ R
as
Q
!8
9 <
<
<
<
. Define
<
(4)
The proof of correctness is through Lucas’ theorem. If . !- B8 then ! . If ! but
." != B&
then !Q ! .
Plugging any of these polynomials into construction 1 gives the following type of graph : Add to B
,
! B where is either a prime or a prime power close to 9 .
if 0) ,
For constructing . -color Ramsey graphs, we define OR representations with . polynomials / I
such that the union of their zero sets if M!NO'P * M FP . We can extend constructions in Equations 1, 2 by taking .
distinct primes. To extend the construction of Equation 3, let 6 < . For . define
4
I
O
< N
(5)
6
I
We can similarly extend Equation 4, we omit the details.
3 Lower Bounds for Prime-power Representations
In this section we prove a lower bound for prime-power representations by symmetric polynomials.
/ 4 S4
/ U4 T Theorem 4 Let Q "
and !Q
"
be symmetric polynomials that represent the OR function
9
] .
on variables. Then 65"DF 65"D)7 ! "
Since a symmetric polynomial on 0-1 inputs is essentially a polynomial in the weight of the input, we can
restate Theorem 4 in terms of integer polynomials. The formal proof is easy and is omitted. While we could
/4S4
, the presence of zero divisors would make it messier to use valuations.
also work with polynomials in
/ Proposition 5 Let !8 $
be univariate polynomials such that for M P ,
4 4 4 6
4 I
QA! P
;I FP or I !QA! P I
!8 P
(6)
Then 65"D) 65"DF7!
$]
9 .
9
The next two Lemmas (6 and 7) develop tools to show degree bounds for such polynomials.
Definition 7 A sequence
4
I
I
of integers is called a greedy sequence if for all ,
<
<
4
I
>
for
"
and
"
< for
. The definition of a greedy sequence can be
Let us define
4 I
4 restated as I
P I
P for )
. Given any set , we can order it elements greedily as follows
to get a greedy sequence: we choose arbitrarily; having chosen ; to be the
we choose I
I
I
4 element that minimizes I
P .
Lemma 6 Let
I
0+<
2
/-
be a greedy sequence. Let Q 4
I
Then 65"D "]
Q
P
46
I
Q
be such that
<
for N ;0 P
.
P ROOF : The proof is by induction on 0 . We will show the converse, namely that if 65"DF 1;0+<
4
I
P
& % '
]
I
I
46
Q
.
P
4 9
4 The base case 0V
is trivial, in this case is constant so it is clear that I P J I P . Assume
I
the property holds for greedy sequences of length 0@< . Given a polynomial Q of degree 0 < , since
is a monic polynomial of degree 0 < , we write Q !Q %
, where !8 is a
I
I
I
polynomial of degree 0+< . Substituting ,
Q
!Q
%
I
I
hence by the ultrametric inequality
I I P
(7)
4
of length
obtained by deleting
To lower bound Q , note that the sequence I is also greedy. Hence applying the inductive hypothesis to Q , we get
I
4
& % ' 4 Q & % ' 4 Q (8)
8 ! for 8
Q . We now lower bound
, hence 8 The last equality follows since I
4 %
. Using the greedy property of the sequence ; ,
I
I
I
4 4 %
%
I
I
I
I
I
% I I I Q I 8 I 4
I
I
!
Q
&%'
PH]
M
4
I
!8
P I
P
4
%
P
9
0
<
!
I
!
P
]
I
9 I
!
PV
N
0
<
Hence we have
P
P
]
I
!
P
I
Since 9 I
I
4
%
!8
I
P
]
9 I
4
P
<
&%'
& % '
4 PH] )M I Q
I
I
I
is a greedy sequence and ! has degree 0+<
4
I
I
I
!Q
10
PV
!
& % '
4 I !Q
P
P
P
I
I
, we get by induction that
9 I
4
Q
P
(9)
(10)
Combining Equations 7, 8, 9, 10 gives the desired result. An example of a greedy sequence is when are consecutive integers. Thus while a degree 0
I
/ 4S
polynomial over
can have several zeroes, the lemma implies that it can have at most 0 consecutive zeroes.
/4S
The intuition for this Lemma is from an algorithm for polynomial interpolation over
by the author [14].
/ 4 S4
Given a set , and values F for of some polynomial in
, the algorithm will output the smallest
degree polynomial that fits the data, provided it sees the elements of in the above greedy order. If the
polynomial is ! on every element but the last, the algorithm is forced to output a polynomial of degree 0+< .
Next we define the notion of a greedy
we use to construct long greedy sequences. Given a
array which
, we use
. -dimensional matrix of dimension 0
0 to denote
N N .
I
I
Definition 8 A . -dimensional matrix of distinct integers
46
I
<
is called a greedy array if
&%')M +
&X1
UZ
N
W
P
W
(11)
We define an ordering of the array indices, which is essentially the reverse lexicographic (revlex) ordering.
Definition 9 Given a . -dimensional integer vectors and , let +
1.243)MUZ
N
W
W
P . Then
if N
.
Note that for a greedy array, the valuation should equal the smallest index where and differ. However to
6 array where
order elements, we look at the largest index where they differ. For example, consider the 6
N ; N J N N =
6 /N 6
I and ! N @
6 < . Thus the array contains N M!N A6 < 'P with
I
I
I
4 numbers indexed by their base-p expansion. Since I N < depends on the smallest digit where N and , this is a
greedy array. The ordering defined above is the usual ordering of integers, it depends on the largest digit where
the expansions differ.
Lemma 7 Ordering elements of a greedy array gives a greedy sequence.
P ROOF : We want to show that for <
, we define the set
For ! ;Z . +N
M
W
4
I
8<
W
W
N
I
W
The indices N N
are unrestricted. Note that the
I
W
show that for every Z , and for ;
I
S
4
8<
X8<
N
I
(12)
&
I
P
s are disjoint and they partition the set M
W
;
I
W
4
I
4
I
S
XX<
Z
I
W
W
W
N
8<
11
N
N
I
'P
X
W
I
W
N-
I
I
P
P . We
(13)
"
;
M 6 + 4 M 6 +
;
M
6
+W N
&
Equation 12 will follow by summing over all Z . Hence consider a fixed Z . Note that if 4 I
8<
Z . Accordingly we partition
Z+<
into A! 5 Z as follows: for !8 ;
+
P
W
then !
,
4 ,
G<
XF
Z since for all
For Z we have I
W
define the sets A! ; Z as follows. For ! ;Z< ,
O M
M
6
4
+
I
W
W
N
+
W
4
I
S
X<
+ +
&
N
W
W
N
W
= N
I
I
W
P
so
W
I
N
W
. Now given
let us
P
P
I
4 , so we have I
XJ<
X]
Z . Since the indices
W
for !8;Z . We now prove Equation 13.
+
4 G K
+ +
I
8<
W
X8<
N
6 + ZN , for N Z it could be that N W
are unrestricted in , we have O
I
W
6
+W N
M
ZN
Unlike for
N N
S
I
4
XX<
&
4 G K
+ +
W
6
I
X <
&
W
]
W
]
+
+
W
Hence the claim follows. A two-dimensional greedy array is a matrix of integers such that elements in the same row are congruent
mod 6 , while elements in distinct rows are not congruent mod 6 . Lemma 7 says that ordering the elements of column-wise gives a greedy sequence.
9
This concludes the algebraic step of the proof. Let us sketch the rest of the proof when 6 < , which
corresponds to the Frankl-Wilson construction (see Figure 1). Define the sets
4
4
M!-P M
M!-P M
9
M A6
<'P
9
M A6
I
!QFP for every Note that I !QA! P they intersect only at ! . We will show that
<'P
+
46
+
I
I
4
QA! P
I
QA! P
];I
4
4
QFPP
FPP
9
! in . Further and partition the set M ;A6
and contain large greedy arrays.
<
'P and
9
1. Arrange M!N6
<'P
in 6
6 grid, each row corresponding to a congruence class mod 6 .
2. Within each row, place elements lying in
elements in which are !-8B&36 before ! .
before those in
3. Sort the rows according to how many elements from
. Since ! lies in
, place all other
they contain.
This reordering is illustrated in Figure 1, the dark line separates and . It is clear that and contain
greedy arrays of size
and of size I respectively9 (indicated by shaded regions) so that
I
6 . From this it follows that +
"] 6 . Also, we can ensure that ! is the last
I
I
4 element of these arrays in the column-wise ordering. So I Q P is minimized at the last element in , hence
by Lemma 7 65"DF ] +
< . Similarly 5"D 7!] <
, which proves the desired bound. Also, +
+ &+
+ &++ +
+ +
12
+ /++ +
0
0
9
Figure 1: Lower bound for 6
and the Frankl-Wilson construction
<
+
+
is minimized when M !- B?6 P , and ! M !- B36 P (or vice versa), the corresponding
polynomials give exactly the Frankl-Wilson construction.
The proof for general is a high dimensional extension of this argument. The next lemma (Lemma 8) says
that any disjoint partition of M!N < 'P into will result in one of the partitions having a greedy array
of size 9 . In fact we prove something stronger, we can
choose the dimensions of the array to be any solution
to Equation 14. We also assume that is of the form 6
I which makes it easier to use induction.
Lemma 8 Let 6 . Let be disjoint sets of integers such that
any positive integers satisfying
I
I
For N .
either
contains a greedy array of size
<
I
M!N 6
I
+ 7+ NF
N contains a greedy array of size M
MN<
7+
N
+
6
+ 7+
N
'P . Given
(14)
P ROOF : The proof is by induction on . (the dimension of the greedy arrays).
When
-
.
,
we
have
disjoint
sets
so
that
M
N
!
<
'P hence ordering
of
and
gives
greedy
arrays
of
size
and
respectively.
Given
< 6 any
, if -
< , then
&]
<
.
Assume that the claim is true up to . < . For !8 N 6 < , we define the following sets
+ +
<
I
or
I
6 I
.
+ 4+ + + . Since
such that
+ +
B&36;P
N P
We define sets
9 N and N similarly. Note that for each N , N and N are disjoint, further N N M!N 6
<
'P . So the induction hypothesis applied to N and N with ; ; I
I I
I
implies that either
contains a greedy array of size
or
contains a greedy array of size
I
I
I I . We define the following sets
)
0
1
M
N
N
N
+ has a greedy array of size I
+ has a greedy array of size I
M
N
13
P
I
I
P
80
+15+ + 07+
+1 +
&%'
+ 07+ + +15+ Since are disjoint and
define a
6 we have either ] or 8] . Assume 8] . We
greedy array of size
as follows. Choose
of size . For each NF
, the N row of I
contains the pre-image of in N of dimension
.
I
I
4
We need to verify that satisfies I < >@ MUZ N P . Given and , if N , then
W
W
, so that N N . Since X &
B 6 so the condition holds. Now assume that N % 8&
B 36 for !8 % 6< . So
and are in the same row, 8<
4
I
X<
X
&%' M +
UZ
N
1
N
W
P
W
&%')M +
UZ
N
W
N
46
?I
Note that
<
6 -
6 <
%O
<
/%' M +
UZ
>
W
W
8<
N
6 -
%O
&
P
P . Hence is a greedy array of the right dimension. W
The next Lemma is the key step in the combinatorial argument. Now we consider sets and
which
intersect only at ! , and we want to produce greedy arrays that end at ! by our ordering. We show that such
arrays exist whose dimensions satisfy Equation 14.
Lemma 9 Partition Lemma: Let M!N 6
6 . Let
I
be sets of integers such that
'P <
M!-P
Then there exist positive integers ; satisfying Equation 14, so that contains a greedy
I
I
array of size
and contains a greedy array of size I , and both and I
contain ! as the last element.
P ROOF : The proof is by induction on . .
M!N < 'P and
M!-P so "
.
When . , we have sets so that We take
. Define to be an ordering of where ! comes last, similarly for
.
Assume that the claim holds up to F
.
For
,
we
define
the
sets
. < ! N 6 < N N N N as
before. Note that
+ 7+
+ +
A! A! + 7+ + +
A! M!N 6
A! M!-P
9
<'P
By induction, there exist
and as above so that A! contains a greedy array of size
I
I
I
I
I . For N 6< we have
and A! contains a greedy array of size I
I
I
9
N N N N M!N 6
)
0
1
M
N
N
N
+ has a greedy array of size I
+ has a greedy array of size I
M
<
'P
Hence applying Lemma 8, either N contains an array of size
I
of size . Again we define the sets
I
I
N
14
I
or
P
I
I
P
A! contains a greedy array
+15+
+ 07+
/ 0
0
M!-P and M!NA6< 'P we have 6 . Order and
Let
. Since
so that ! is the last element. We define a greedy
array of size
as follows. For each N
,
I
the N row of contains the pre-image of in N of dimension
. Similarly we define
I
I
where the N row contains the pre-image of in N . The proof that these are greedy arrays follows
Lemma 8. They both contain ! as the last element by induction. 0
We now complete the proof of Theorem 4.
P ROOF OF T HEOREM
4:
6 . We can choose so that Assume that 6
I M!-P
1
M
6
6
6 and 6
I
I <
I
46
I <
46
QA! P
;I
QA! P
];I
4
4
<
I FPP
FPP
. Define the sets
2
Applying Lemma 9 implies that and contain greedy arrays and of size
and I
I
respectively where and satisfy Equation 14. Applying Lemma 7, by ordering we get a greedy sequence
4 4 . By the definition of set , I QA! P
M ; :!-P in of length 0( I
Q
P for N ;0 . So by
I
I
] < .
Lemma 6 65"D) "
Similarly we get a greedy sequence of length in ending in ! . Note that by Equation 6, and
4 4 ! implies I !QA! P I
!Q
FP . So by Lemma 6, 65"D)7 ! ] < . By Equation 14, ] 6 for
. <
and ]
. Hence
I
I
M!-P
6+ M 6+ 65"D) 65"D 7! ]
6
]
9
For the Frankl-Wilson construction where 6
I
>
9
, we get 65"DF 65"DF7! ] 6
<
which is tight. 4 Lower Bounds for Prime Representations
In this section we prove a lower bound for prime representations using symmetric polynomials.
/5O
4 / #
Theorem 10 Let Q "
and !Q
"
be symmetric polynomials that represent the OR function
on variables. Then 65"D) /65"D 7 !"
] -"! .
Note that this requires 65"D) / 65"D 7 !
] but if 5"D F
! , then it is easy to show that 65"D 7 ! F
, so
this case is not interesting. The hard case of this theorem is when 6 and ' are fast-growing functions of , as in
Alon’s construction. To handle this case, we prove a partition lemma (Lemma 11) which says that taking 6 and
' large does not help.
/ 4
/ #
68' . Let
Definition 10 Let 6 ' be distinct primes, let and
. We say that
if . We say and cover if every M P is covered by or .
B ?6
/ 4 and / , the number of
+ 4+ + + + 7++ + which can be much larger than
+ 7 ++ + needs to be => .
cover , then ,+ 7
+O ,+ +O > .
/ 4
/ #
68' , we can cover by taking If and
. Given
elements in M A68' P that are covered by or is
'
6<
. The partition lemma states that to cover the first integers however,
+ 4++ +
Lemma 11 Partition Lemma: If
/ 4
and
/
#
is covered by
15
#
9
,!"/.01#"*234
+ !"%)"%("*!'
!#"$$!"%!"&#('
Figure 2: Proof of Prop 12
+ 7+
+ 4+
+ 7+
,+ +
in the product lets us ignore the case when K ! . Let us sketch the idea
Using rather than
#
/ #
behind the proof of the Partition Lemma. Let ' . Assume that to begin with, we have
and
#
M '- '- ' P . It is clear that and cover , however U
. One could try and
reduce
by removing elements from it. We want to show that this results in an increase in
. Removing
/ #
#
N-
from results in the numbers M N N ' N > < ' P being uncovered. Call this set <N . The
6 ' and they are congruent mod ' , hence the CRT implies they cannot also
various elements of < N are less than 8
be congruent mod 6 . But the problem is for N ,there could be considerable overlap between the residues
of < N and < - mod 6 . Hence it is not clear that removing many elements from does actually cause
/14
to increase. However, by suitably reordering the elements of , we show that every element removed from
causes the size of to increase by at least . In fact Figure 2 shows that
could increase by just . This is
sufficient to prove the Partition Lemma.
4
#
/ 4
4 7
B 36
B ?6 P .
Set 65 8
and 95 #)7 . Given set of integers, define 8&
to be the set M 8&
+ +
,+ 7+
>
'
. If
and
cover and
+ 7+
Proposition 12 Let )]
+ +
1'<
then
+ 4+
+ 6
+ 7+ 65 7
]
+ 4+
#
<
.
#
/ #
P ROOF : Note that since )];' , ] . Let denote the complement of in , so !(
. For each N , take
#
#
<N to be the first numbers in M P congruent to N B36 . In other words, <A! MU' '" ' P
#
and for N !N N F M N N '-N) > < ' P . Let
; : < )
6
<
N
, then is not covered by so it must be covered by . We want to lower bound the size of ) B36 .
/ 4
Let us reorder the set
as M!N/'- '- 6(< ' P (this is a reordering since ' A6 ). It sends = B36
#
= B36 . This map sends <A! B?6 to MU'-;
' P and the set <N B?6 to
to % - ' such that % - '
#
#
the interval M% N '-A% N 5 '-;A%N < ' P of length for N+ ! . None of these intervals contain
! , since that would give M ; P such that N B.' and !-8B&36 . Such an is not covered by
or . Each interval < N 8&
B 36 begins at a distinct point % N . Sorting the intervals by their starting points, it
#
#
/ 4
follows that the union of such intervals of length contains at least "< elements of . /
Figure 2 illustrates this argument for @
6 = / '= . Here * M!NO'P .
II
P ROOF OF L EMMA 11:
6 ,6
;
' and ' separately. The non-trivial case is when ' .
We consider the cases If 16
1. Let 6
;'
. Numbers M ; P lie in distinct congruence classes mod 6 and ' . Hence
+ 7+O+ +
,+ 4+O
&]
2. Let 6
O
,+ +"
>
' . The numbers M A6;P lie in distinct congruence classes modulo 6 and ' . Hence
6 and O O 6 . This proves the claim if 6 so let 6 .
4
4
6H<
for 8
6 . There are numbers in each congruence class mod 6 . Thus numbers are not covered by and have to be covered by . Since ' , they lie in distinct congruence
4
4
] 6 hence 6 ] we get
classes mod ' . Hence =] . Using the fact that )
+ 4+ + +
Let + 4+
&]
,+ 7+
+ +
,+ 4O+ ,+ +
>
,+ +"
+ +
>
>
1' <
3. Let ;' . By Prop. 12, if
#
Hence for ? 6< we have
,+ 7+O
>
, then
+ 7+
#
6]
,+ +
8
6< >
4
6 ]
+ 7+
. Since
<
'<
]
4
-
#
FA
6=< , we get 6=< #
.
We will show that this is lower bounded by . By differentiating, this bound is minimized at one of the
extreme values of , so it suffices to check the bound is at least 9 for those values. When $ ,
',<
When 5 7
'
4 '
'
>
#
6<
'<
'
]
'<
6 '<
6 '
7>
One of '< 6 6 and >N6F4' is at least : If '
'
',< 6F 6 6
N6
'
7>
6 , >N6F4'
6
. If '
6 , then '< 6F 6
>
.
/ 4 We now proceed to the algebraic step of the proof. Every symmetric polynomial Q V computes
/ 4
a symmetric function . Let 6 . Every polynomial R also computes
/54
a function . The following equivalence between the two kinds of polynomials is given by Theorem
2.4 of [10].
Proposition 13 The functions that can be computed by symmetric polynomials Q =
;
.
than 6 I are the functions which can be computed by polynomials Q
/4O
of degree less
>
For each variable , let 65"D denote the degree of
in . If is the largest index so that 65"D) !
6 . This gives a bound with an error factor of 6 . By defining an appropriate weighted
then 6 65"D) degree of , we can make the correspondence exact.
17
; / 4 , the degree of a monomial with 6 is
6 . The degree of denoted "D is the maximum degree over all monomials.
"D) 6 "D) .
Note that if is the largest index such that "D >! then "D) 8
/4
there is a unique polynomial Q R that
Lemma 14 Given a symmetric polynomial Q /4
computes the same function and vice versa. This correspondence preserves the degree.
% /14
P ROOF : Given a symmetric multilinear polynomial Q of degree , write it as Q G K . By Lucas’ Theorem
On at 0-1 input N
B ?6
. Thus computes the same function as
Further the polynomial
has degree 6
; R
% 8B ?6
Definition 11 Given defined as 65"D =
0
65
5
H<
5
0
;65
;65
"
0
&
and they have the same degree.
with 6 < form a basis for
To prove the other direction, observe that the monomials /5O
4 with degree at most 6< in each . Further writing a polynomial in this basis
polynomials in
. Hence
6 be the degree of the monomial does not change the degree as defined above. Let given Q
with degree 0 , one can write
Q ; % I
6 By Lucas’ theorem, this computes the same function as the polynomial Q
% -
. For M!N; P , let denote the base 6 and base ' expansions of . For
L
I '
/ 4 QL
;L =
L
;L let Q denote the polynomial applied to the base 6 expansion of . As
consequence of Lemma 14, to prove Theorem 10 it suffices to prove the following Proposition.
Proposition 15 Let Q
For / 4
"
L
,
JU65"D) ! "]
I
;L
QA!
Further "5 D Q
.
and
!
$8B&36
$
/#%
I
<
I
and 8
! A! !, B?6
or Q
! U
be polynomials such that
$ B.'
!, B&V'
] in . This implies 65"D) +
] 6 W and Q ,
P ROOF : Let Z denote the largest index such that 65"DF L =
W
QL
;L . Similarly let [ be the largest index so that 65"D I ] . Then 65"DF ! ] ' Y and !Q W
Y
!8
I I . Hence 65"D 65"D ! ] 6 W ' Y . This proves the desired bound for " !R6 W ' Y . So we may
Y
assume that )
] "!R6 W ' Y .
6 W I ' Y I , since if 6 W
I ' Y I , then L L ! and I ;I ! so
Also 68W
I
' Y
W
I QA!N
18
:! Y
B&36
.
!Q
6
which contradicts the hypothesis. Let I
W
' Y
54
Let us consider inputs of the form
# 6
# B ?6 . Similarly
;
and
as A!N ;:!N
, and <
"DF # 6 and "DF 5
"D) 1
L
65
L
W
' Y
H
I
<
W
65
/ 4
We define
and
. So by Lemma 11,
Since 65"D)L
65"DF ! ];'
Y
W
' Y
165
A! H
#
/
#
I
Y
I S%# T
7
!QA!N;:!
68' .
W ' Y
or
to be the ! sets of
W
#
65"D
W
]
"!R6
W ' Y
65"D I
68W
$>
hence 6
Y
. Since 65"D)
]
54
S # T
and
cover
" !
W ' Y
]
' Y
#
(15)
respectively. By equation 15
Y
! The second inequality uses the fact that )]
!- B('
65"D I Y
J
B.'
and <
, this implies that 65"DFL
#
and <A! !- B36
$ B36
$ B36
65"D
#
where !8
. Observe that this implies L L
!
W /I 4 I
! and I
6 W B.' . Define polynomials +
I /# Y
Y as <
!8A!N;:!N 6 W . This implies 65"D H
H
;'< . Note that
"5 D)L
/65"D)I ]
Y
65"D I ,
>
]
6
W
5"D L
W
and
" !
7 > I
.
5 Ramsey Graphs based on Set Intersections
The constructions of Frankl-Wilson, Alon and Grolmusz use a coloring scheme based on the size of set intersections. In this section we show that these can be constructed from certain polynomials that are closely related to
OR polynomials. These polynomials are also used by Grolmusz [18] and Kutin [19] to give simple constructions
of set systems with restricted intersections mod 6.
0
Definition 12 The weight- function is a partial function defined on M!NO'P for ] as follows
The function is undefined for
."
! if
."
if
."
@ .
Note that on variables is simply the NAND function. We now define polynomial representations of
. We give an extension of Definition 2, a similar extension holds for Definition 4
/ O
4 / #
$
and !8
"
V represent the Definition 13 Polynomials Q "
function on variables if
0
Q
Q
!,8B&?6
!,8B&?6
and 8
! !, B&V'
or
!8
!, B.'
if
if . . The degree of the representation is 01V243 65"DF /65"D 7! .
0
0
and !8 represent Assume that Q with degree 0 for some ] . Define
I
I
by
substituting
Q
and !8 to be polynomial
<? for when N? and setting
I
I
! for N] . It is easy to verify that and ! represent
OR
on
variables
with degree at most 0 . Further
if and ! were symmetric polynomials, then so are and ! . Thus lower bounds for OR representations
19
imply lower bounds for representations. In particular our lower bounds for OR representations rule out
representations of with symmetric polynomials of degree 9 .
Conversely one can construct degree 0 representations of from degree 0 symmetric polynomials repre
senting OR on variables. We do not know if a similar statement is true for asymmetric polynomials.
Lemma 16 A degree 0 representation of OR on variables using symmetric polynomials gives a degree
representation of on variables for all ] ,
0
P ROOF : Let !Q be symmetric polynomials of degree at most 0 on variables that represent
OR.
Replace each by < and multi-linearize. It is easy to show that the resulting polynomials ! represent on variables. Further, they are symmetric multilinear polynomials of degree 0 , hence we can
write them as
I
;
We obtain new polynomials
I
;
0
Z
and
Z
!
I
!
by replacing
I
;
0
!
I
I
;
I
;
[
by
0
0
I
[
.
;
I
I
0
;
Since the
value of a symmetric function on a !NO -input depends only on the weight of the input, one can show
that and ! represent on variables. We can use the 78 9 OR representations to construct representations of . We give a construction of
explicit Ramsey graphs from representations of .
Construction 2 Ramsey Graph 7A5/B from0 representations of .
consists of vectors M !NO'P
of weight .
<
A <
! , add to B .
If Q Theorem 17 Given a0 degree 0 representation of the weight-n function on M!NO'P
and #
.
0
, the graph 0
has vertices
P ROOF : Assume we have prime representation of . We associate a polynomial with each vertex . Given I I 0 , let ! if I ! and if I . Set
so that @ Q I
0
(
Q ;
and multi-linearize. Using an argument like in Theorem 3, we can show that this
I
/#4
gives a polynomial
representation of over . Since the s are multilinear polynomials of degree 0 , we
0
/#
get 1
. Similarly we bound by representing over .
/<4 S
and get a similar bound. For prime-power representations of OR, we can represent and over
The OR representation of Equation 1 gives the following construction due to Alon [2]. Let 68' < . The vertices are all subsets of @ of size . Add to B if < B&9 6 .
9
The OR representation of Equation 3 gives the Frankl-Wilson construction: Let 6 < . The
vertices are all subsets of @ of size . Add edge to B if + K <
&
B 36 . This construction can
be extended to "
colors using the polynomials constructed in Equation 5.
. ]
9
+ 3 +
+ : +
20
Construction 3 Extending the Frankl-Wilson construction to . colors.
Take <
6 <
6 I . Vertices are all subsets of .
4
<
Edges are colored M!N .G< 'P . Edge is given color I + .
+ 7
+
9
The OR representation of Equation 2 gives the following graph 87A /B . Let <
. The
X
<$ B8
. In fact the graph obtained
vertices are all subsets of @ of size . Add to B if is the same as Grolmusz. To show this, we first present his construction, following the simplified exposition of
Grolmusz himself [18] and Kutin [19].
< . The BBR polynomials give the following representation of .
Let + +
Q
<
!8 <
/2 to be the polynomial obtained by combining these polynomials using the CRT. It follows
Define "
.
that $ B8L when ." and is divisible by 2 or 3 when ."
We can view as an integer polynomial with coefficients in M!N P . By repeating each monomial
sufficiently many times, we can write
The elements of the universe are the monomials. If is repeated % times
in , then there are % elements
0
of weight , the set < consists of
in the universe, one for each occurance of . For each M!NO'P
monomials that evaluate to on . One can verify that this system has restricted intersections mod 6 since
< < .
The vertices of the graph are the sets < . We add edge < if < !- B8 .
We wish to show that this graph is the same as the graph constructed above. We identify ;A with < )AQ . In , we add an edge between < and < if @ !,8B&Q . By the CRT, this
! ,
B Q . By Lucas’ theorem, this happens if . < BQ
implies that . But this is
precisely when is an edge of .
This equivalence can also be seen from Kutin’s construction [19]. While Grolmusz’s set system construction
is an important result, our approach seems to be simpler for the purpose of Ramsey graph construction. One
can interpret the bounds on clique and independent set size as coming from an extension of the modular RayChaudhuri-Wilson theorem to prime powers, which we prove below (Theorem 18).
+1
9
+
P
+
3 6
+
5.1 Set Systems with restricted Intersections modulo Prime Powers
+1 +
1+ +
Definition 14 A set system C
48B&.' but
so that
M
4
on is said to have restricted intersections mod ' if there exists B.' .
/
#
+ 6+
For a fixed modulus ' , we study the problem of how large C can be as a function of . When 'V 6 is a
prime, the non-uniform modular Ray-Chaudhuri Wilson theorem proved by Deza, Frankl and Singhi [5] gives a
bound of
C 6 <
+ 6+
When
4 S '
I
+ .+ is not a prime power, Grolmusz shows a lower bound of EHG IK [17]. We give a near-tight bound of
9 for the prime power case. This improves the bound of
due to Babai et al. [6]. Previously
21
+ +
stronger bounds than ours were known for the special case when 6 W < i.e. when all set sizes are congruent
to 8&
B 6 W for some (see theorems 5.30 and 7.18 in the book by Babai and Frankl [5]). To prove our result,
/#4 S
/ 4
we use the fact that every function from
to
can be written as a polynomial.
Theorem 18 Let C
be a set system with restricted intersections modulo 6
P ROOF : We construct a univariate integer-valued polynomial
QF
"
R
B&36
B?6
W
!= B&36
B?6
W
W
. Then
+ 6+N 4 I .
S
C
of degree 6
W
<
such that
By Lucas’ theorem,
6
<
6
5
6 W <
<
W
<
W
<
6
W
<$8B&36 W
6
W
<$8B&36 W
$
B&6
!= B&6
F
B&6
!= B&6
Q
Set
$
6 W <
6 W <
B36
W
B36
W
By Lemma 3.1, of [6] this implies the desired bound. We sketch the argument below.
to denote the incidence vector of set . Let ; Q
We will use
I
linearize. It is easy to show that
B ?6
,+1 ++
!
B ?6
Using this one can show that the polynomials are linearly independent over
1
polynomials in variables of degree 6
W
<
N
,
N
and multi-
R
. Since they are multilinear
, the bound follows. 6 Discussion and Open Problems
Following the breakthrough of Barak et al. [8], the algebraic construction described here are no longer the best
constructions known. However, the appeal of these constructions is their simplicity and elegance, together with
the fact that they are very explicit. So we believe that it is important to resolve the question of whether this
approach can beat the Frankl-Wilson bound. As we have seen, this problem is intimately linked to well-studied
questions in complexity theory.
Lower Bounds for Asymmetric Polynomials:
The question of whether there are lower degree weak representations of the OR function mod 6 has been open
for a while. This work raises the question of whether low degree OR representations exist for our definition.
Better upper bounds would give better Ramsey graphs. Lower bounds for prime representations will imply
lower bounds for weak representations mod 6. Prime-power representations are exciting from the lower bound
viewpoint since they have not been studied previously and might turn out to be easier to work with. The =A@CBD
lower bound of Barrington and Tardos citebar-tar applies to both kinds of representations.
22
Both our lower bounds for symmetric polynomials follow a similar scheme: we characterize the zero-sets
of low-degree symmetric polynomials and then show that there is no good partition of the hypercube. A natural
question is whether such a scheme could extend to the general case. The first step would be to give a of
characterization of zero-sets of low degree polynomials. Motivated by this we pose the following problems:
/ 4 4
1. Given
M!NO'P * , let 65"D 5 denote the smallest degree of a polynomial in
which is ! at
4
every point in but not at the origin. Give a lower bound on 65"D 5 .
/ 4 4
2. Given
M!NO'PO , let 65"D 5 denote the smallest degree of a polynomial in
which is ! over
4
but not at every point in M!NO'P . Give a lower bound on 65"D 5 .
Note that both these quantities are easy to compute, since they involve checking whether a system of equations is feasible.
We are looking for a combinatorial lower bound, perhaps analogous to Lemma 6. The latter
quantity 0 5 is closely related to the notion of the degree of a subset studied by Smolensky with a view
towards proving circuit lower bounds [21]. The main difference is that he requires the zero-set to be exactly the
set .
Limitations to Constructions based on Distances:
We have shown that using symmetric polynomials in out construction, current techniques cannot give better
bounds on #
. Note that for the constructions of Alon, Frankl-Wilson and Grolmusz, this technique
gives tight bounds.This raises the question: do constructions based on symmetric polynomials contain either a
large clique or independent set?
Using a symmetric polynomial in our construction gives a graph where edges are added between vertices
based on the Hamming distance between them. More formally, let M P . The graph is defined
as follows: The vertex set is M!NO'P' . We add to B if 0 . Is it true that for every choice of ,
contains a large clique or independent set?
Similarly in Construction 2, symmetric polynomials give graphs where the vertices are sets and edges are
added based on intersection sizes. Do such graphs always contain large cliques or independent sets?
Acknowledgments
I thank Dick Lipton for numerous interesting discussions on this subject. I thank Vince Grolmusz for comments
on an early version. I also thank Boaz Barak, T.S. Jayram, Ravi Kumar, Anup Rao, Benny Sudakov, Prasad
Tetali and Avi Wigderson for helpful comments.
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