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SUMMARY:The Multicolour Ramsey Number of a Long Odd Cycle - Jozef Skokan (
 LSE)
DTSTART:20171012T133000Z
DTEND:20171012T143000Z
UID:TALK88951@talks.cam.ac.uk
CONTACT:Andrew Thomason
DESCRIPTION:For a graph $G$\, the $k$-colour Ramsey number $R_k(G)$ is the
  least integer $N$ such that every $k$-colouring of the edges of the compl
 ete graph $K_N$ contains a monochromatic copy of $G$. Let $C_n$ denote the
  cycle on $n$ vertices. We show that for fixed $k\\ge 3$ and $n$ odd and s
 ufficiently large\, $$R_k(C_n)=2^{k-1}(n-1)+1.$$ This generalises a result
  of Kohayakawa\, Simonovits and Skokan and resolves a conjecture of Bondy 
 and Erd\\H{o}s for large $n$. We also establish a surprising correspondenc
 e between extremal $k$-colourings for this problem and perfect matchings i
 n the hypercube $Q_k$. This allows us to in fact prove a stability-type ge
 neralisation of the above. The proof is analytic in nature\, the first ste
 p of which is to use the Regularity Lemma to relate this\nproblem in Ramse
 y theory to one in convex optimisation.\n\nThis is joint work with Matthew
  Jenssen.
LOCATION:MR12
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