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 $t \geqslant 4$, there are $K_t$-abundant graphs of chromatic...
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Main Authors | , , , |
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Format | Journal Article |
Language | English |
Published |
27.10.2023
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Subjects | |
Online Access | Get full text |
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Summary: | 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 $t \geqslant 4$, there are $K_t$-abundant graphs of
chromatic number $t$. Using similar methods, we also extend work of Ruzsa by
proving that a set $\mathcal{A} \subset \{1,\dots,N\}$ which avoids solutions
with distinct integers to an equation of genus at least two has size
$\mathcal{O}(\sqrt{N})$. The best previous bound was $N^{1 - o(1)}$ 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. |
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DOI: | 10.48550/arxiv.2310.18202 |