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Grid-free linear hypergraphs via Cayley-Bacharach

Cosmin Pohoata

Abstract

We give a new construction showing that for every $r\ge 3$, there exists an $r$-uniform linear hypergraph on $n$ vertices with $Θ_r(n^2)$ edges and no copy of the $r\times r$ grid. This complements the works of Füredi--Ruszinkó, Glock--Joos--Kim--Kühn--Lichev, Delcourt--Postle for $r \geq 4$, as well as the subsequent constructions of Gishboliner--Shapira and Solymosi for the case $r=3$.

Grid-free linear hypergraphs via Cayley-Bacharach

Abstract

We give a new construction showing that for every , there exists an -uniform linear hypergraph on vertices with edges and no copy of the grid. This complements the works of Füredi--Ruszinkó, Glock--Joos--Kim--Kühn--Lichev, Delcourt--Postle for , as well as the subsequent constructions of Gishboliner--Shapira and Solymosi for the case .
Paper Structure (8 sections, 8 theorems, 39 equations, 1 figure)

This paper contains 8 sections, 8 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Construction
  3. Proof of Theorem \ref{['thm:main']}
  4. Punctured intersections
  5. A degree budget lemma
  6. Application to the construction $H_{r,q}$ from Theorem \ref{['thm:main']}
  7. A denser model: transversals of $r$ parallel lines
  8. Concluding remarks

Key Result

Theorem 1.1

For every $r \geq 3$ and every sufficiently large odd prime power $q$, there exists an $r$-uniform linear hypergraph $H_{r,q}$ with no $r\times r$ grid, with Equivalently, for $n=rq$,

Figures (1)

  • Figure 1: The punctured intersection $P_3(T)$ with $T=\{(3,3)\}$: the intersection $R_3\cap C_3$ is removed and replaced by private vertices $a_{33}\in R_3$ and $b_{33}\in C_3$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • proof
  • proof
  • proof
  • Lemma 3.1: Cayley-Bacharach
  • proof : Proof of Theorem \ref{['thm:main']}
  • Definition 4.1
  • Definition 4.2
  • Lemma 4.3
  • ...and 8 more