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, 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 ⊂{1, ^..., N} which avoids solutions with distinct integers to an equation of genus at least two has size O (√N). The best previous bound was N^1-0(1)
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.