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- W4317553055 abstract "The triangle removal states that if $G$ contains $varepsilon n^2$ edge-disjoint triangles, then $G$ contains $delta(varepsilon)n^3$ triangles. Unfortunately, there are no sensible bounds on the order of growth of $delta(varepsilon)$, and at any rate, it is known that $delta(varepsilon)$ is not polynomial in $varepsilon$. Csaba recently obtained an asymmetric variant of the triangle removal, stating that if $G$ contains $varepsilon n^2$ edge-disjoint triangles, then $G$ contains $2^{-mathrm{poly}(1/varepsilon)}cdot n^5$ copies of $C_5$. To this end, he devised a new variant of Szemer'edi's regularity lemma. We obtain the following results: - We first give a regularity-free proof of Csaba's theorem, which improves the number of copies of $C_5$ to the optimal number $mathrm{poly}(varepsilon)cdot n^5$. - We say that $H$ is $K_3$-abundant if every graph containing $varepsilon n^2$ edge-disjoint triangles has $mathrm{poly}(varepsilon)cdot n^{|V(H)|}$ copies of $H$. It is easy to see that a $K_3$-abundant graph must be triangle-free and tripartite. Given our first result, it is natural to ask if all triangle-free tripartite graphs are $K_3$-abundant. Our second result is that assuming a well-known conjecture of Ruzsa in additive number theory, the answer to this question is negative. Our proofs use a mix of combinatorial, number-theoretic, probabilistic, and Ramsey-type arguments." @default.
- W4317553055 created "2023-01-20" @default.
- W4317553055 creator A5042083332 @default.
- W4317553055 creator A5084893376 @default.
- W4317553055 creator A5089978763 @default.
- W4317553055 date "2023-01-18" @default.
- W4317553055 modified "2023-09-27" @default.
- W4317553055 title "An efficient asymmetric removal lemma and its limitations" @default.
- W4317553055 doi "https://doi.org/10.48550/arxiv.2301.07693" @default.
- W4317553055 hasPublicationYear "2023" @default.
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