Department of
Applied
Mathematics

Ramsey numbers for restricted colorings

Martin Balko

Abstract

We discuss Ramsey numbers of ordered hypergraphs in restricted colorings. That is, we ask for the minimum positive integer N such that every coloring c, which avoids certain forbidden ordered subconfigurations, of the ordered complete r-uniform hypergraph on N vertices contains a monochromatic copy of a given ordered r-uniform hypergraph.

We consider several types of restricted colorings and we show that some of them admit natural geometric interpretations that yield new results for various extremal problems in discrete geometry. In particular, we mention connections between estimating Ramsey numbers of monotone paths in so-called monotone colorings and higher-order Erdős--Szekeres theorems. We also address a problem posed by Eliáš and Matoušek and by Moshkovitz and Shapira about the growth rate of Ramsey numbers of monotone paths in so-called transitive colorings.