Abundance: Asymmetric graph removal lemmas and integer solutions to linear equations

Abstract

We prove that a large family of pairs of graphs satisfy a polynomial dependence in asymmetric graph removal lemmas. In particular, we give an unexpected answer to a question of Gishboliner, Shapira and Wigderson by showing that for every $t⩾4$, there are $K_t$-abundant graphs of chromatic number $t$. Using similar methods, we also extend work of Ruzsa by proving that a set $A\subset \{1,\cdots ,N\}$ which avoids solutions with distinct integers to an equation of genus at least two has size $O\left(\sqrt{N}\right)$. The best previous bound was $N^{1-o\left(1\right)}$ and the exponent of $1/2$ is best possible in such a result. Finally, we investigate the relationship between polynomial dependencies in asymmetric removal lemmas and the problem of avoiding integer solutions to equations. The results suggest a potentially deep correspondence. Many open questions remain.