The 3-colored Ramsey Number of Even Cycles
Fabrı́cio Benevides
1.
1
Jozef Skokan2
The result
Denote by R(L, L, L) the minimum integer N such that any 3-coloring of the edges of the
complete graph with N vertices KN contains a monochromatic copy of a graph L. Bondy and
Erdős [1] conjectured that for an odd n-cycle Cn , R(Cn , Cn , Cn ) = 4n − 3 for n > 3. This is
sharp if true.
Luczak [4] proved that if n is odd, then R(Cn , Cn , Cn ) = 4n + o(n), as n → ∞. Kohayakawa,
Simonovits and Skokan proved that the exact Bondy-Erdős conjecture holds for sufficiently large
values of n. Figaj and Luczak [2] determined an assintotic result for the ‘complementary’ case
where the cycles are even: they showed that for n even R(Cn , Cn , Cn ) = 2n + o(n) (Actually
their result is much stronger than that because the cycles may be of slightly different sizes).
Slightly later, independently, Gyárfás, Ruszinkó, Sárközy, and Szemerédi [3] proved a similar
but more precise result for paths: there exists an n0 such that for n > n0 R(Pn , Pn , Pn ) = 2n − 1
or 2n − 2 depending on the parity of n.
Now we prove the following theorem.
Theorem 1. There is n0 such that for n > n0 , n even we have R(Cn , Cn , Cn ) = 2n.
2.
Overview of the proof
To prove the lower bound (i.e., R(Cn , Cn , Cn ) > 2n − 1), we just need to look to the following
coloring of a graph with 2n − 1 vertices.
Coloring 1 (ECM AX ). Take 4 groups of vertices, A, B, C, D, of n/2 − 1 vertices each (recall
that n is even). Color the edges inside each group arbitrarily, the edges in E(A, B) ∪ E(C, D)
by Red, the edges in E(A, D) ∪ E(B, C) by Green, and the edges in E(A, C) ∪ E(B, D) by Blue.
Now add one new vertex r and let all edges from r to the other 2n − 4 vertices be Red. In the
same way add a vertex g and let all the 2n − 3 edges at this vertex be Green. And then add a
vertex b and let all the 2n − 2 edges at this vertex be Blue.
Claim 2. For any n even, coloring ECM AX do not contain any monochromatic Cn .
Our proof fot the upper bound mostly follows the proof-line of Gyárfás, Ruszinkó, G. Sárközy,
and Szemerédi [3], but we strength some of their lemma to find cycles instead of just paths. We
need to look at two different colorings. We found convenient to consider multi-3-colorings instead
of 3-colorings. In a multi-3-coloring of a graph G, some of the edges can be assigned more than
∗
one color. We denote GR the subgraph induced by the edges of G that are colored only with
color Red (and similarly for the other colors).
1
Instituto de Matemática e Estatı́stica, Universidade de São Paulo, Rua do Matão 1010, 05508–090 São Paulo,
Brazil. This project has received support from FAPESP process 05/52494-0
2
Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, United
Kingdom and Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Benevides, Skokan
October 28, 2006
2
The first extremal coloring is just a relaxation of ECM AX .
Coloring 2 (EC1 (with parameter α)). There exists a partition V (G) = A ∪ B ∪ C ∪ D such
that
• |A|, |B|, |C|, |D| ≥ (1 − α) |V (G)|
4 ,
∗
∗
∗
∗
• The bipartite graphs (A × B) ∩ GR , (C × D) ∩ GR , (A × D) ∩ GG , (B × C) ∩ GG ,
∗
∗
(A × C) ∩ GB , (B × D) ∩ GB are all (1 − α)-dense.
Coloring 3 (EC2 (with parameter α)). There exists a partition V (G) = A ∪ B ∪ C ∪ D such
that
• |A|, |B|, |C|, |D| ≥ (1 − α) |V (G)|
4 ,
∗
∗
∗
• The bipartite graphs (A × B) ∩ GR , ((A ∪ B) × C) ∩ GG , ((A ∪ B) × D) ∩ GB are all
(1 − α)-dense.
The main tool to prove the upper bound in our result is the following stability theorem
proved in [3].
Theorem 3 (Lemma 1 in [3]). For every sufficiently small α there exists positive reals η3 , 3
and positive integer n3 such that for every n ≥ n3 the following holds: for < 3 if a (1−)-dense
graph Gn is 3-multi-colored then we have one of the following cases.
Case 1 Gn contains a monochromatic connected matching of size at least ( 14 + η3 )n edges.
Case 2 This is an Extremal Coloring 1 (EC1 ) with parameter α/2.
Case 3 This is an Extremal Coloring 2 (EC2 ) with parameter α/2.
The strategy to prove the upper bound is the following. We first apply the edge-colored
version of the Regularity Lemma to a three-colored K2n . We build the so called Reduced
Graph, the graph whose vertices are associated to the clusters and whose edges are associated
to -regular pairs. The edges of the reduced graph are multicolored with the colors of density at
least α4 between the clusters. Then we apply the above lemma for this graph and in each case
we prove that we can find a monochromatic cycle in the original graph.
References
[1] J. A. Bondy and P. Erdős, Ramsey numbers for cycles in graphs, J. Combinatorial Theory
Ser. B 14 (1973), 46–54. MR 0317991 (47 #6540)
[2] A. Figaj and T. Luczak, The Ramsey number for a triple of long even cycles, J. Combin.
Theory Ser. B (2006), In Press.
[3] A. Gyárfás, M. Ruszinkó, G. N. Sárközi, and E. Szemerédi, Three-color Ramsey number
for paths, submitted, 2005.
[4] T. Luczak, R(Cn , Cn , Cn ) ≤ (4 + o(1))n, J. Combin. Theory Ser. B 75 (1999), no. 2, 174–
187. MR 1676887 (2000b:05096)