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arXiv · 2104.11626

Removal lemmas and approximate homomorphisms

Jacob Fox·Yufei Zhao
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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.

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Jacob Fox, Yufei Zhao. 2021-04-23. Removal lemmas and approximate homomorphisms. https://arxiv.org/abs/2104.11626

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