Erdos-Hajnal-type theorems in hypergraphs

The Erdos-Hajnal conjecture states that if a graph on n vertices is H-free, that is, it does not contain an induced copy of a given graph H, then it must contain either a clique or an independent set of size n^{d(H)}, where d(H) > 0 depends only on the graph H. Except for a few special cases, this conjecture remains wide open. However, it is known that a H-free graph must contain a complete or empty bipartite graph with parts of polynomial size. We prove an analogue of this result for 3-uniform hypergraphs, showing that if a 3-uniform hypergraph on n vertices is H-free, for any given H, then it must contain a complete or empty tripartite subgraph with parts of order c(log n)^{1/2 + d(H)}, where d(H) > 0 depends only on H. This improves on the bound of c(log n)^{1/2}, which holds in all 3-uniform hypergraphs, and, up to the value of the constant d(H), is best possible. We also prove that, for k > 3, no analogue of the standard Erdos-Hajnal conjecture can hold in k-uniform hypergraphs. That is, there are k-uniform hypergraphs H and sequences of H-free hypergraphs which do not contain cliques or independent sets of size appreciably larger than one would normally expect.Comment: 15 page

Problems and memories

I state some open problems coming from joint work with Paul Erd\H{o}sComment: This is a paper form of the talk I gave on July 5, 2013 at the centennial conference in Budapest to honor Paul Erd\H{o}

Local colourings and monochromatic partitions in complete bipartite graphs

We show that for any $2$-local colouring of the edges of the balanced complete bipartite graph $K_{n,n}$, its vertices can be covered with at most~$3$ disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite $r$-locally coloured graph with $O(r^2)$ disjoint monochromatic cycles.\\ We also determine the $2$-local bipartite Ramsey number of a path almost exactly: Every $2$-local colouring of the edges of $K_{n,n}$ contains a monochromatic path on $n$ vertices.Comment: 18 page