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

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$t \geqslant 4$, there are Kt$K_t$‐abundant graphs of chromatic number t$t$. Using similar methods, we also extend work of Ruzsa by proving that a set A⊂{1,⋯,N}$\mathcal {A}\subset \lbrace 1,\dots,N \rbrace$ which avoids solutions with distinct integers to an equation of genus at least two has size O(N)$\mathcal {O}(\sqrt {N})$. The best previous bound was N1−o(1)$N^{1 - o(1)}$ and the exponent of 1/2$1/2$ 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. [ABSTRACT FROM AUTHOR]