Preliminaries
Main Result
The Four Folkman Graphs
Explicit Construction of Small Folkman Graphs
Michael Dairyko
Spectral Graph Theory
Final Presentation
April 17, 2017
Dairyko
Explicit Construction of Small Folkman Graphs
Preliminaries
Main Result
The Four Folkman Graphs
Notation
Rado’s Arrow
Consider two graphs G and H. Then G → (H)p is the statement
that if the edges of G are p-colored, then there exists a
monochromatic subgraph of G isomorphic to H.
K6 → (K3 )2
K5 6→ (K3 )2
Dairyko
Explicit Construction of Small Folkman Graphs
Preliminaries
Main Result
The Four Folkman Graphs
Motivation
Conjecture: Erdős and Hajnal (1967)
For each p there exists a graph G , containing no K4 , which has the
property that G → (K3 )p .
The case when p = 2 was proven by the following theorem:
Folkman’s Theorem (1970)
For any k2 > k1 ≥ 3, there exist a Kk2 -free graph G with
G → (Kk1 )2 .
Thus, any K4 -free graph G with G → (K3 )2 is a Folkman Graph.
Dairyko
Explicit Construction of Small Folkman Graphs
Preliminaries
Main Result
The Four Folkman Graphs
Motivation
Conjecture: Erdős and Hajnal (1967)
For each p there exists a graph G , containing no K4 , which has the
property that G → (K3 )p .
Folkman’s theorem was generalized as:
Nes̆etr̆il - Rödl’s Theorem (1976)
For p ≥ 2 and any k2 > k1 ≥ 3, there exist a Kk2 -free graph G
with G → (Kk1 )p .
Thus, there exist a K4 -free graph G with G → (K3 )p .
Dairyko
Explicit Construction of Small Folkman Graphs
Preliminaries
Main Result
The Four Folkman Graphs
Motivation
Question
For any k1 < k2 and any p ≥ 2, what is the smallest integer
f (p, k1 , k2 ) = n such that there is a Kk2 -free graph G on n vertices
satisfying G → (Kk1 )p .
• Graham (1968) proved that f (2, 3, 6) = 8 by showing that
K8 \ C5 → (K3 )2 .
Dairyko
Explicit Construction of Small Folkman Graphs
Preliminaries
Main Result
The Four Folkman Graphs
Motivation
Question
For any k1 < k2 and any p ≥ 2, what is the smallest integer
f (p, k1 , k2 ) = n such that there is a Kk2 -free graph G on n vertices
satisfying G → (Kk1 )p .
• Graham (1968) proved that f (2, 3, 6) = 8 by showing that
K8 \ C5 → (K3 )2 .
Dairyko
Explicit Construction of Small Folkman Graphs
Preliminaries
Main Result
The Four Folkman Graphs
Motivation
Question
For any k1 < k2 and any p ≥ 2, what is the smallest integer
f (p, k1 , k2 ) = n such that there is a Kk2 -free graph G on n vertices
satisfying G → (Kk1 )p .
• Graham (1968) proved that f (2, 3, 6) = 8 by showing that
K8 \ C5 → (K3 )2 .
Dairyko
Explicit Construction of Small Folkman Graphs
Preliminaries
Main Result
The Four Folkman Graphs
Motivation
Question
For any k1 < k2 and any p ≥ 2, what is the smallest integer
f (p, k1 , k2 ) = n such that there is a Kk2 -free graph G on n vertices
satisfying G → (Kk1 )p .
• Graham (1968) proved that f (2, 3, 6) = 8 by showing that
K8 \ C5 → (K3 )2 .
Dairyko
Explicit Construction of Small Folkman Graphs
Preliminaries
Main Result
The Four Folkman Graphs
Motivation
Question
For any k1 < k2 and any p ≥ 2, what is the smallest integer
f (p, k1 , k2 ) = n such that there is a Kk2 -free graph G on n vertices
satisfying G → (Kk1 )p .
• Graham (1968) proved that f (2, 3, 6) = 8 by showing that
K8 \ C5 → (K3 )2 .
Dairyko
Explicit Construction of Small Folkman Graphs
Preliminaries
Main Result
The Four Folkman Graphs
Motivation
K5 -free graphs G with G → (K3 )2 : n = f(2,3,5)
Authors
n
Graham, Spencer (1971)
Irving (1973)
Khadzhiivanov and Nenov (1979)
Nenov (1981)
Piwakowski, Radziszowski, and Urbański (1998)
Dairyko
≤ 23
≤ 18
≤ 16
≤ 15
≥ 15
Explicit Construction of Small Folkman Graphs
Preliminaries
Main Result
The Four Folkman Graphs
Motivation
Bounds for f(2,3,4)
• Both upper bounds of Folkman and of Nes̆etr̆il and Rödl for
f (2, 3, 4) are extremely large.
• Frankl and Rödl (1986) showed f (2, 3, 4) ≤ 7 × 1011 .
Erdös $$ prize
Erdös set a prize of $100 for the challenge f (2, 3, 4) ≤ 1010 .
• Spencer (1988) showed that f (2, 3, 4) ≤ 3 × 109 .
Again, Erdös set a prize of $100 for the challenge f (2, 3, 4) ≤ 106 .
• This paper claimed the reward in 2008!
Dairyko
Explicit Construction of Small Folkman Graphs
Preliminaries
Main Result
The Four Folkman Graphs
$100 Theorem
Theorem: LU (2007)
f (2, 3, 4) ≤ 9697.
Proof Sketch
• Use spectral analysis to establish a sufficient condition for
G → (K3 )2 .
• Examine a special class of graphs and find the four ”small”
Folkman graphs of size 9697, 30193, 33121, 57401.
• Get that cash money.
Dairyko
Explicit Construction of Small Folkman Graphs
Preliminaries
Main Result
The Four Folkman Graphs
The Graph L(m, s)
Let gcd(m, s) = 1 and let n be the smallest integer such that
sn ≡ 1
mod m.
L(m, s)
The graph L(m, s) is the circulant graph on m vertices generated
by S = {s i mod m : i = 0, 1, . . . , n − 1}.
Spectrum Of Circulant Graphs
The eigenvalues of the adjacency matrix for the circulant graph
generated by S ⊂ Zn are
X
2πis
cos
,
n
s∈S
for i = 0, 1, . . . , n − 1.
Dairyko
Explicit Construction of Small Folkman Graphs
Preliminaries
Main Result
The Four Folkman Graphs
The Graph L(m, s)
Lemma
H, the unique local graph of L(m, s) is isomorphic to a circulant
graph of order n.
proof sketch:
• V (H) = S, E (H) = {xy : x ∈ S, y ∈ S, x − y ∈ S}.
• Define the bijection f : Zn → S such that f (i) = s i mod n.
• Since f (i + j) = f (i)f (j), f is a group isomorphism.
• Define T ⊂ Zn such that T = {i : f (i) − 1 ∈ S}.
• Let H 0 be the circulant graph generated by (Zn , T ).
• f is in fact a group homomorphism, mapping H 0 to H.
Dairyko
Explicit Construction of Small Folkman Graphs
Preliminaries
Main Result
The Four Folkman Graphs
Proof Sketch Of Main Theorem
• H is a local graph of L(m, s)
• A is the adjacency matrix for H.
• Let σ = λλmin where λmin , λmax ∈ spec(A).
max
Results From Computation
If σ > − 13 , then
L(m, s) → (K3 )2 .
Via a computer algorithm in Maple,
L(9697, 4), L(30193, 53), L(33121, 2), and L(57401, 7)
are Folkman graphs.
Dairyko
Explicit Construction of Small Folkman Graphs
Preliminaries
Main Result
The Four Folkman Graphs
Works Cited
L. Lu, Explicit construction of small Folkman graphs, SIAM J.
Discrete Math., 21 (2008), pp. 1053 -1060.
Dairyko
Explicit Construction of Small Folkman Graphs