Chaya Keller: On multicolor Ramsey numbers and subset-coloring of hypergraphs

Abstract: The multicolor hypergraph Ramsey number $R_k(s,r)$ is the minimal n, such that in any k-coloring of all r-element subsets of [n], there is a subset of size s, all whose r-subsets are monochromatic. We present a new "stepping-up lemma" for $R_k(s,r)$: If $R_k(s,r)>n$, then $R_{k+3}(s+1,r+1)>2^n$. Using the lemma, we improve some known lower bounds on multicolor hypergraph Ramsey numbers. Furthermore, given a hypergraph $H=(V,E)$, we consider the Ramsey-like problem of coloring all r-subsets of $V$ such that no hyperedge of size >r is monochromatic. We provide upper and lower bounds on the number of colors necessary in terms of the chromatic number $\chi \left(H\right)$. In particular, we show that this number is $O\left(lo{g}^{(r-1)}\right(r\chi \left(H\right))+r)$, where $lo{g}^{\left(m\right)}$ denotes m-fold logarithm. Joint work with Bruno Jartoux, Shakhar Smorodinsky, and Yelena Yuditsky.