Removal lemmas and approximate homomorphisms

TL;DR

This paper explores the quantitative links between the triangle removal lemma and its variants, establishing bounds on approximate homomorphisms and their growth rates, with implications for graph theory and finite field analogues.

Contribution

It introduces the triangle-free lemma with bounds on approximate homomorphisms and shows these bounds grow faster than any polynomial in inverse epsilon, extending results to general graphs and finite fields.

Findings

Least possible M grows faster than exponential in any polynomial of epsilon^{-1}

Bounds are close to optimal for finite field analogues

Results apply to arbitrary graphs and their arithmetic analogues

Abstract

We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each $\epsilon>0$ there exists $M$ such that every triangle-free graph $G$ has an $\epsilon$-approximate homomorphism to a triangle-free graph $F$ on at most $M$ vertices (here an $\epsilon$-approximate homomorphism is a map $V(G) \to V(F)$ where all but at most $\epsilon |V(G)|^2$ edges of $G$ are mapped to edges of $F$). One consequence of our results is that the least possible $M$ in the triangle-free lemma grows faster than exponential in any polynomial in $\epsilon^{-1}$. We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal.