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Polynomial removal lemma for ordered matchings

Lior Gishboliner, Borna Šimić

TL;DR

This paper proves a polynomial removal lemma for ordered $s$-uniform matchings on $t$ vertices: for any $\varepsilon > 0$, an ordered $s$-uniform hypergraph $G$ on $n$ vertices that is $\varepsilon$-far from being $H$-free contains at least $\left(\frac{\varepsilon}{C}\right)^C n^t$ copies of $H$, where $C = C(t)$. The authors develop a novel nested-partition and cleaning framework to respect vertex order, then lift a single ordered copy of $H$ found in a cleaned stage to many copies in the original graph, yielding a polynomial-type bound despite the order constraints. They extend the approach to general $s$-uniform hypergraphs and discuss tightness aspects via a lower-bound proposition, which suggests $C(t) \ge t/s$ for ordered matchings. The results advance understanding of polynomial bounds in ordered removal lemmas and provide techniques potentially applicable to broader ordered forest families and hypergraphs.

Abstract

We prove that for every ordered matching $H$ on $t$ vertices, if an ordered $n$-vertex graph $G$ is $\varepsilon$-far from being $H$-free, then $G$ contains $\text{poly}(\varepsilon) n^t$ copies of $H$. This proves a special case of a conjecture of Tomon and the first author. We also generalize this statement to uniform hypergraphs.

Polynomial removal lemma for ordered matchings

TL;DR

This paper proves a polynomial removal lemma for ordered -uniform matchings on vertices: for any , an ordered -uniform hypergraph on vertices that is -far from being -free contains at least copies of , where . The authors develop a novel nested-partition and cleaning framework to respect vertex order, then lift a single ordered copy of found in a cleaned stage to many copies in the original graph, yielding a polynomial-type bound despite the order constraints. They extend the approach to general -uniform hypergraphs and discuss tightness aspects via a lower-bound proposition, which suggests for ordered matchings. The results advance understanding of polynomial bounds in ordered removal lemmas and provide techniques potentially applicable to broader ordered forest families and hypergraphs.

Abstract

We prove that for every ordered matching on vertices, if an ordered -vertex graph is -far from being -free, then contains copies of . This proves a special case of a conjecture of Tomon and the first author. We also generalize this statement to uniform hypergraphs.
Paper Structure (3 sections, 4 theorems, 8 equations)

This paper contains 3 sections, 4 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main']}
  3. Concluding remarks

Key Result

Theorem 1

For every $t \geq s \geq 2$, there exists $C = C(t)$ such that the following holds. Let $\varepsilon > 0$, let $H$ be an ordered $s$-uniform matching on $t$ vertices, and let $G$ be an ordered $s$-uniform hypergraph on $n$ vertices. If $G$ is $\varepsilon$-far from being $H$-free, then $G$ contains

Theorems & Definitions (11)

  • Theorem 1
  • Lemma 1
  • proof : Proof of Lemma \ref{['lem:main']}
  • Claim 1
  • proof
  • Claim 2
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • ...and 1 more