On the Border of Finite and Infinite
Lajos Soukup
Alfréd Rényi Institute of Mathematics
Hungarian Academy of Sciences
Horizon of Combinatorics, 2006
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
1 / 38
Outline
1
Splitting Antichains (P. L. Erdős, –)
2
Quasi Kernels and Quasi Sinks ( P. L. Erd ős, A. Hajnal, –)
3
Multiway Cuts
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
2 / 38
ts
Property B
Definition
A family H of subsets of a set A has property B iff there is a
partition(X , Y ) of A such that H ∩ X 6= ∅ 6= H ∩ Y for each H ∈ H.
H
hH, 1i
hH, 1i
a
0iA
a
1i
X
0i
1i
XH
Y
hH, 0i
hH, 0i
Soukup, L (Rényi Institute)
Y
Q(H) = h A ∪ (H × {0, 1}) , ≤i.
h0, Hi ≤ a ≤ h1, Hi iff a ∈ H.
If ∅ ∈
/ H then A is a maximal
antichain
A = X ∪∗ Y
(X , Y ) witnesses that H has
property B iff Q(H) = X ↓ ∪ Y ↑
On the Border of Finite and Infinite
HC 2006
3 / 38
Splitting antichains
Finite posets
Definition
Let P = hP, ≤i be a poset, A ⊂ P be a maximal antichain.
A ⊂ P splits iff A has a partition A = B ∪∗ C such that P = B ↑ ∪ C ↓ .
nts
A
B
C
B
C
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
4 / 38
Property B
Theorem (Lovász, 1979)
≥2
such that |H 0 ∩ H 00 | 6= 1 for each
If A is a finite set and H⊂ A
2
{H, H 0 } ∈ H then H has property-B.
H
hH 00 , 1i
Theorem (Lovász, reformulated)
b
Let A be a finite set, H ⊂ P(A),
∅∈
/ H, s. t. if
hH 0 , 0i ≤ a ≤ hH 00 , 1ithen there is
b 6= a such that
hH 0 , 0i ≤ b ≤ hH 00 , 1i. Then A splits
in Q(H).
ts
A
a
0i
1iH
b
a
hH 0 , 0i
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
5 / 38
Property B
z
Definition
Let P be a poset and y ∈ P. y is a
cut point iff ∃x, z ∈ P s.t.
x <P y <P z and
[x, z] = [x, y] ∪ [y,
z]. replacements
PSfrag
A subset A ⊂ P is cut-free if it
does not contain cut points.
z
y
x
x
Theorem (Lovász, reformulated)
Let A be a finite set and H ⊂ P(A), ∅ ∈
/ H. If A is cut-free (in Q(H))
then A splits.
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
6 / 38
Splitting antichains
Finite posets
Theorem (P. L Erdős - Niall Graham (1993))
In a finite Boolean lattice every max. antichain splits.
Theorem
If P is a finite poset and A is a maximal antichain then the question
“Does A split?” is NP-complete.
Theorem (Ahlswede, P. L. Erdős, N. Graham(1995))
In a finite poset every cut-free maximal antichain splits.
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
7 / 38
Splitting antichains
Infinite posets
Theorem (Ahlswede, Khachatrian )
There is a maximal, (infinite) non-splitting antichain A in divisor poset
of the square-free positive integers.
D E
<ω
divisor poset of of the square-free positive integers = ω
,⊂
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
8 / 38
Splitting antichains
Infinite posets
Definition
<ω
A poset P is loose iff for each x ∈ P and F ∈ P
if x ∈
/ F ↑ then
there is y ∈ x ↑ \ {x} such that y 6∈ F l .
y
nts F
x
y
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
9 / 38
Splitting antichains
Infinite posets
Fact
D E
<ω
ω
, ⊂ is loose.
Theorem (P. L. Erdős, – )
A countable, cut-free, loose poset P = hP, ≤i contains a maximal
infinite non-splitting antichain A.
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
10 / 38
Splitting antichains
Infinite posets
Theorem (P. L. Erdős)
In a cut-free poset every finite maximal antichains split.
Theorem
If P is a poset, A is a cut-free maximal antichain such that |x l ∩ A| < ω
for all x ∈ P then A splits.
No proofs with Gödel Compactness Theorem!
If P is cut-free, Q ⊂ P then Q is not necessarily cut-free
Theorem (P. L. Erdős, – )
Assume that Pis a countable poset such that both P and P −1 are
loose. Then P contains a maximal antichain which splits.
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
11 / 38
Splitting antichains
Infinite posets
Example
There is a “non-trivial” infinite cut-free poset s.t. every maximal
antichain splits.
Problem
Is there a countable cut-free poset without splitting maximal
antichains?
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
12 / 38
Splitting antichains
Uncountable posets
Deep set-theory, independence, ...
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
13 / 38
Quasi Kernels and Quasi Sinks
Theorem (Chvatal, Lovász)
Every finite digraph (i.e. directed graph) contains an independent set
A such that for each point v there is a path of length at most 2 from
some point of A to v.
rag replacements
G
A
a
v
A
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
14 / 38
In and Out
Definition
Assume that G = (V , E) is a digraph, A ⊂ V and n ∈ N. Let
Inn (A) = {v ∈ V : there is a path of length at most n
which leads from v to some points of A}, and
Outn (A) = {v ∈ V : there is a path of length at most n
which leads from some points of A to v }.
Definition
Let G = (V , E) be a digraph.
An independent set A is a quasi-kernel if and only if V = Out 2 (A).
An independent set B is a quasi-sink if and only if V = In 2 (B).
Theorem (Chvatal, Lovász)
Every finite digraph G = (V , E) contains a quasi-kernel (quasi-sink).
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
15 / 38
Finite 6= Infinite
ragPlain
replacements
generalization fails even for infinite tournaments:
the tournament (Z, <) is a counterexample.
−1
Soukup, L (Rényi Institute)
0
1
On the Border of Finite and Infinite
2
HC 2006
16 / 38
The original problem
(Z, <) does not have quasi kernel, but Z = Out1 (1) ∪ In1 (0).
Problem
Is it true that for each directed graph G = (V , E) there are disjoint,
independent subsets A and B of V such that V = Out2 (A) ∪ In2 (B).
cements
A
B
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
17 / 38
Hunting for counterexamples
Definition
G = (V , E) is a digraph, n, k ∈ N:
G ∈ Ink ⇐⇒ ∃ an independent set A ⊂ V s. t. V = In k (A),
G ∈ Outn ⇐⇒ ∃ an independent set B ⊂ V s.t. V = Outn (B).
G ∈ Ink -Outn ⇐⇒ ∃ partition (V1 , V2 ) of V s.t G[V1 ] ∈ Ink and
G[V2 ] ∈ Outn .
Theorem (Chvatal, Lovász)
Every finite digraphs is in Out 2 .
Theorem
Every tournament is either in Out2 or in In1 -Out1 .
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
18 / 38
Hunting for counterexamples
Theorem
If G = (V , E) is a digraph and In1 (x) is finite for each x ∈ V then
G ∈ Out2 .
Theorem
If the chromatic number of G is finite then G ∈ Out 2 .
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
19 / 38
Hunting for counterexamples
Definition
If G = (V , E) is a digraph define the undirected complement of the
e = (V , E)
e if and only if (x, y) ∈
e as follows: {x, y} ∈ E
graph, G
/ E and
(y, x) ∈
/ E.
Theorem
e for some n ≥ 2 then
Let G = (V , E) be a directed graph. If Kn 6⊂ G
e is finite then
G ∈ In2 -Out2 . Especially, if the chromatic number of G
G ∈ In2 -Out2 .
Theorem
e is locally finite then
If G = (V , E) is a digraph such that G
G ∈ In2 -Out2 .
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
20 / 38
A theorem and a new conjecture
Theorem
For each directed graph G = (V , E) there are disjoint, independent
subsets A and B of V such that V = Out2 (A) ∪ In2 (B).
In the positive theorems we obtained G ∈ In 2 -Out2 !
Conjecture
Every directed graph is in In 2 -Out2 .
Problem
Find an infinite digraph G s. t. G ∈
/ In1 -Out2 .
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
21 / 38
Structure theorems for tournaments
PSfrag replacements
PSfrag replacements
Let T∞ = (N, E), where (x, y) is an edge if and only if y = x + 1 or
y + 1 < x.
G∞
0
G∞
1
2
3
G∞
0
G∞
1
2
3
T∞ ∈
/ Out2 , T∞ ∈
/ Outn for n ∈ N
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
22 / 38
Structure
PSfrag replacements
theorems for tournaments
Let G∞ = (N, E), as follows: (x, y) is an edge if and only if x ≥ y + 1.
0
1
2
3
G∞
G∞
Theorem
For an infinite tournament T = (V , E) the followings are equivalent:
(i) T ∈
/ Out3 ,
(ii) T ∈
/ Outn for each n ≥ 3,
(iii) there is a surjective homomorphism ϕ : T → G ∞ .
Theorem
There is a tournament T ∈ Out3 \ Out2 .
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
23 / 38
Multiway cuts
nts
s1
s2
s3
t
s3 }
=2
Multiway Cut Problem
Fix a graph G = (V , E) and a subset S of vertices called terminals. A
multiway cut is a set of edges whose removal disconnects each
terminal from the others. The multiway cut problem is to find the
minimal size of a multiway cut denoted by π G,S .
s2
s1
t
S = {s1 , s2 , s3 }
πG,S = 2
s3
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
24 / 38
Multiway cuts
Definition
~ = (V , E) is a directed graph and A, B ⊂ V let λ( G,
~ A, B) be the
If G
maximal number of edge-disjoint directed paths from some element
of A into some element of B.
Definition
If G = (V , E) is a finite graph, S ⊂ V , and
~ is obtained from G by an orientation of
G
PSfrag replacements
the edges, then let
X
~ S − s, s)
νG,S
=
λ(G,
~
s1
s2
t
S = {s1 , s2 , s3 }
s∈S
~ S − s3 , s3 ) = 1, λ(G,
~ S − s2 , s2 ) = 1,
λ(G,
~ S − s1 , s1 ) = 0, ν ~ = 2
πG,S = 2
λ(G,
G,S
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
s3
HC 2006
25 / 38
Multiway cuts
Theorem (P. L. Erdős, A. Frank, L. Székely)
~ is obtained from G by an
If G = (V , E) is a finite graph, S ⊂ V , and G
orientation of the edges, then νG,S
≤ πG,S .
~
Proof.
Fix a multiway cut F ⊂ PSfrag
E and replacements
for s ∈ S let
Ps be a family of edge-disjoint directed
paths from some element of S − s into s.
For each P ∈ Ps let ep be the last element
of P ∩ F in P.
eP
Then eP 6= eP 0 provided P 6= P 0 .
P
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
s1
s2
x
z
P
eP
y
t
HC 2006
26 / 38
Multiway cuts
Theorem (E. Dahjhaus, D. S. Johson, C. H. Papadimitriou, P.D.
Seymout, M Yannakakis)
The multiway cut problem is NP-complete.
Special case:
G − S is a tree.
nts
S
S
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
27 / 38
Multiway cuts
Theorem (P. L. Erdős, L. Székely)
If G = (V , E) is a finite graph, S ⊂ V such that G − S is tree, then
max νG,S
= πG,S .
~
~
G
~ of G.
where the maximum is taken over all orientations G
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
28 / 38
Multiway cuts
Theorem (P. L. Erdős, A. Frank, L. Székely, reformulated)
If G = (V , E) is a finite graph, S ⊂ V such that G − S is tree, then
~ of G, and for each s ∈ S there is an
there is an orientation G
~ and for each P ∈ Ps
edge-disjoint family Ps of (S − s, s)-paths in G
we can pick an edge eP ∈ P such that
{eP : P ∈ Ps for some s ∈ S}
is a multiway cut (in G for S).
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
29 / 38
Multiway cuts
Theorem (–)
If G = (V , E) is a graph, S ⊂ V such that G − S is tree without
~ of G, and for each s ∈ S
infinite paths, then there is an orientation G
~ and for
there is an edge-disjoint family Ps of (S − s, s)-paths in G
each P ∈ Ps we can pick an edge eP ∈ P such that
{eP : P ∈ Ps for some s ∈ S}
is a multiway cut (in G for S).
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
30 / 38
Multiway cuts
Proposition
Let G = (V , E) be a finite directed graph, and A, B ⊂ V s.t
(1) in(a) = 0 and out(a) = 1 for each a ∈ A,
(2) in(b) = 1 and out(b) = 0 for each b ∈ B,
(3) in(x) ≤ out(x) for each x ∈ V \ (A ∪ b).
Then there is a family P of edge-disjoint A-B-paths s .t. P covers A.
A
B
nts
V \ (A ∪ B)
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
31 / 38
Multiway cuts
Infinite case
Theorem
Let G = (V , E) be a directed graph which does not contain infinite
directed path, and let A, B ⊂ V s.t
(1) in(a) = 0 and out(a) = 1 for each a ∈ A,
(2) in(b) = 1 and out(b) = 0 for each b ∈ B,
(3) in(x) ≤ out(x) for each x ∈ V \ (A ∪ B).
Then there is a family P of edge-disjoint A-B-paths s .t. P covers A.
Proof.
G is countable: easy induction: if P is an A-B-path then G − P satisfies
(1)–(3)
G is uncountable may got stuck at some point
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
32 / 38
Multiway cuts
Infinite case
aω
nts
a2
b2
a1
b1
v
b0
B) a0
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
33 / 38
Multiway cuts
Uncountable case
Inductive construction, but using the right enumeration
Elementary submodels
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
34 / 38
From Infinite to Finite
Unfriendly Partitions
Definition
Let G = (V , E) be a graph. A partition (A, B) of V is called unfriendly
iff every vertex has at least as many neighbor in the other class as in
its own.
Observation
Every finite graph has an unfriendly partition.
Theorem (Shelah)
There is an uncountable graph without an unfriendly partition.
Unfriendly Partition Conjecture
Every countable graph has an unfriendly partition.
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
35 / 38
From Infinite to Finite
Unfriendly Partitions
Fact
Every locally finite graph has an unfriendly partition.
Fact
If G = (V , E) is countable and every v ∈ V has infinite degree then G
has an unfriendly partition.
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
36 / 38
From Infinite to Finite
Unfriendly Partitions
Question
Let G be a finite graph, and a and b are vertices such that
10
dG (a, b) ≥ 1010 . Is there an unfriendly partition of (A, B) of G such
that a ∈ A and b ∈ B?
Answer
No, V. Bonifaci gave counterexample.
Question
Is it true that for each n ∈ N there is f (n) ∈ N such that for each finite
graph G if deg(x) ≤ n for each vertices, and a and b are vertices such
that dG (a, b) ≥ f (n). then there is an unfriendly partition of (A, B) of G
such that a ∈ A and b ∈ B?
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
37 / 38
Summary
Many finite problems have infinite counterparts.
Similar, but not the same.
Deep set-theory is not a must.
Soukup, L (Rényi Institute)
On the Border of Finite and Infinite
HC 2006
38 / 38