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Caps and Wickets

Jakob Führer, Jozsef Solymosi

TL;DR

This work examines the Turán number $ex_L(n,W)$ for the wicket in 3-uniform linear hypergraphs, situating the problem at the crossroads of extremal hypergraph theory and additive combinatorics. It develops a cap-set–based Ruzsa–Szemerédi construction that embeds a three-partite hypergraph in $\mathbb{F}_3^{n+1}$ using a cap-set $S$ to define edges, and reduces wicket configurations to a pair of linear relations constrained by cap-set properties. By applying the Lovász Local Lemma to a random $(1/k)$-coloring of edges and selecting the largest color class, the authors obtain a wicket-free hypergraph with at least $3^n|S|/k$ edges, yielding the explicit bound $ex_L(m,W) \\ge m^{1.544}$ and linking upper bounds on $ex_L$ to cap-set size bounds in $\mathbb{F}_3^n$. The paper also discusses consequences, variants with parameter $\alpha$ and Eisenstein integers, and open questions (e.g., Gowers–Long conjecture and cap-set size growth), highlighting a deep connection between extremal hypergraph theory and additive combinatorics. This approach provides a new route to tighten lower bounds on $ex_L(n,W)$ and suggests that progress on cap-set bounds could directly impact extremal problems for wickets.

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 Turán number of wickets using estimates on cap sets. We also show that this problem is closely connected to important questions in additive combinatorics.

Caps and Wickets

TL;DR

This work examines the Turán number for the wicket in 3-uniform linear hypergraphs, situating the problem at the crossroads of extremal hypergraph theory and additive combinatorics. It develops a cap-set–based Ruzsa–Szemerédi construction that embeds a three-partite hypergraph in using a cap-set to define edges, and reduces wicket configurations to a pair of linear relations constrained by cap-set properties. By applying the Lovász Local Lemma to a random -coloring of edges and selecting the largest color class, the authors obtain a wicket-free hypergraph with at least edges, yielding the explicit bound and linking upper bounds on to cap-set size bounds in . The paper also discusses consequences, variants with parameter and Eisenstein integers, and open questions (e.g., Gowers–Long conjecture and cap-set size growth), highlighting a deep connection between extremal hypergraph theory and additive combinatorics. This approach provides a new route to tighten lower bounds on and suggests that progress on cap-set bounds could directly impact extremal problems for wickets.

Abstract

Let 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 point matrix. In this note, we give a new lower bound on the Turán number of wickets using estimates on cap sets. We also show that this problem is closely connected to important questions in additive combinatorics.
Paper Structure (4 sections, 1 theorem, 9 equations, 6 figures)

This paper contains 4 sections, 1 theorem, 9 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Caps
  3. Related Problems
  4. Acknowledgements

Key Result

Corollary 1

Any upper bound of the form $ex_L(m,W)\leq m^{2-c}$ would lead to an upper bound of $3^{\frac{4}{3}(1-c)n}$ for the size of a cap set in $\mathbb{F}_3^n$.

Figures (6)

  • Figure 1: The wicket is drawn as a three-partite hypergraph. If we add the edge spanned by vertices $D,E,F$, it is isomorphic to a $3\times 3$ grid.
  • Figure 2: After eliminating $x$ and $y$ we get $s+t=2h$
  • Figure 3: In the wicket, $G=x+s=y+t, D=x+2s=z+2v, F=y+u=z+v$ and $C=x+2w=y+2u$.
  • Figure 4: Every wicket in $\mathcal{H}^{(3)}(A,B,C)$ lies in a $2$-dimensional affine subspace.
  • Figure 5: $e, f$ and $e',f'$ share a vertex
  • ...and 1 more figures

Theorems & Definitions (5)

  • Definition
  • Definition
  • Claim 1
  • proof
  • Corollary