An efficient asymmetric removal lemma and its limitations
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 $\...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000689/type/journal_article |
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| Summary: | 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
$2^{-\operatorname {\mathrm {poly}}(1/\varepsilon )}\cdot n^5$
copies of
$C_5$
. To this end, he devised a new variant of Szemerédi’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
$\operatorname {\mathrm {poly}}(\varepsilon )\cdot n^5$
.
•
We say that H is
$K_3$
-abundant if every graph containing
$\varepsilon n^2$
edge-disjoint triangles has
$\operatorname {\mathrm {poly}}(\varepsilon )\cdot n^{\lvert V(H)\rvert }$
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.
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| ISSN: | 2050-5094 |