Caps and Wickets

TL;DR

This paper establishes a new lower bound on the Turán number for wickets, a specific hypergraph structure, by leveraging cap set estimates and explores its connections to additive combinatorics.

Contribution

It introduces a novel lower bound for the Turán number of wickets and links this problem to key questions in additive combinatorics.

Findings

New lower bound on Turán number for wickets

Connection between hypergraph problems and additive combinatorics

Use of cap set estimates in hypergraph extremal problems

Abstract

Let $H_n^{(3)}$ be a 3-uniform linear hypergraph, i.e. any two edges have at most one vertex common. A special hypergraph, {\em wicket}, is formed by three rows and two columns of a $3 \times 3$ point matrix. In this note, we give a new lower bound on the Tur\'an number of wickets using estimates on cap sets. We also show that this problem is closely connected to important questions in additive combinatorics.