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

## Description of video

 Date: 5/7/21 Speaker : Keller Chaya

## Keywords

Abstract: The multicolor hypergraph Ramsey number ${R}_{k}\left(s,r\right)$ 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 ${R}_{k}\left(s,r\right)$: If ${R}_{k}\left(s,r\right)>n$, then ${R}_{k+3}\left(s+1,r+1\right)>{2}^{n}$. Using the lemma, we improve some known lower bounds on multicolor hypergraph Ramsey numbers. Furthermore, given a hypergraph $H=\left(V,E\right)$, 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 $O\left(lo{g}^{\left(r-1\right)}\left(r\chi \left(H\right)\right)+r\right)$, where $lo{g}^{\left(m\right)}$ denotes m-fold logarithm. Joint work with Bruno Jartoux, Shakhar Smorodinsky, and Yelena Yuditsky.

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