Hypergraph Ramsey numbers: Triangles versus cliques

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Abstract

A celebrated result in Ramsey Theory states that the order of magnitude of the triangle-complete graph Ramsey numbers R(3,t) is t2/logt. In this paper, we consider an analogue of this problem for uniform hypergraphs. A triangle is a hypergraph consisting of edges e,f,g such that |ef|=|fg|=|ge|=1 and efg=. For all r2, let R(C3,Ktr) be the smallest positive integer n such that in every red–blue coloring of the edges of the complete r-uniform hypergraph Knr, there exists a red triangle or a blue Ktr. We show that there exist constants a,br>0 such that for all t3,at32(logt)34R(C3,Kt3)b3t32 and for r4t32(logt)34+o(1)R(C3,Ktr)brt32. This determines up to a logarithmic factor the order of magnitude of R(C3,Ktr). We conjecture that R(C3,Ktr)=o(t3/2) for all r3. We also study a generalization to hypergraphs of cycle-complete graph Ramsey numbers R(Ck,Kt) and a connection to r3(N), the maximum size of a set of integers in {1,2,,N} not containing a three-term arithmetic progression.

Keywords

Ramsey number
Hypergraph
Loose triangle
Independent set

Cited by (0)

1

Research of this author is supported in part by NSF grant DMS-0965587 and by grants 12-01-00448-a and 12-01-00631 of the Russian Foundation for Basic Research.

2

Research supported in part by NSF grants DMS 0653946 and DMS 0969092.

3

Research supported by NSF grant DMS 1101489.