The Erdos-Szekeres problem and an induced Ramsey question

TL;DR

This paper explores a new induced Ramsey problem for hypergraphs motivated by the Erdos-Szekeres conjecture, establishing exponential lower bounds for the minimum size of hypergraphs guaranteeing large independent sets under specific edge-induction constraints.

Contribution

It introduces a novel induced Ramsey problem for hypergraphs related to the Erdos-Szekeres conjecture and provides exponential lower bounds for the minimal hypergraph size.

Findings

Established that g_k(n) > 2^{cn^{k-4}} for some constant c(k).

Linked the problem to the Erdos-Szekeres convex polytope conjecture.

Initiated the study of induced Ramsey properties in hypergraphs with specific edge patterns.

Abstract

Motivated by the Erdos-Szekeres convex polytope conjecture in $R^d$, we initiate the study of the following induced Ramsey problem for hypergraphs. Given integers $ n > k \geq 5$, what is the minimum integer $g_k(n)$ such that any $k$-uniform hypergraph on $g_k(n)$ vertices with the property that any set of $k + 1$ vertices induces 0, 2, or 4 edges, contains an independent set of size $n$. Our main result shows that $g_k(n) > 2^{cn^{k-4}}$, where $c = c(k)$.