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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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000689/type/journal_article |
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| author | Lior Gishboliner Asaf Shapira Yuval Wigderson |
| author_facet | Lior Gishboliner Asaf Shapira Yuval Wigderson |
| author_sort | Lior Gishboliner |
| collection | DOAJ |
| description | 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.
|
| format | Article |
| id | doaj-art-5d4867e778644355bb56101ef6f97ac5 |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-5d4867e778644355bb56101ef6f97ac52025-02-10T12:03:33ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.68An efficient asymmetric removal lemma and its limitationsLior Gishboliner0https://orcid.org/0000-0003-0688-8111Asaf Shapira1https://orcid.org/0000-0001-9902-0164Yuval Wigderson2https://orcid.org/0000-0001-5909-9250Department of Mathematics, University of Toronto, Toronto, Canada; E-mail:School of Mathematics, Tel Aviv University, Tel Aviv, IsraelInstitute for Theoretical Studies, ETH Zürich, Zürich, Switzerland; E-mail: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. https://www.cambridge.org/core/product/identifier/S2050509424000689/type/journal_article05C3511B75 |
| spellingShingle | Lior Gishboliner Asaf Shapira Yuval Wigderson An efficient asymmetric removal lemma and its limitations Forum of Mathematics, Sigma 05C35 11B75 |
| title | An efficient asymmetric removal lemma and its limitations |
| title_full | An efficient asymmetric removal lemma and its limitations |
| title_fullStr | An efficient asymmetric removal lemma and its limitations |
| title_full_unstemmed | An efficient asymmetric removal lemma and its limitations |
| title_short | An efficient asymmetric removal lemma and its limitations |
| title_sort | efficient asymmetric removal lemma and its limitations |
| topic | 05C35 11B75 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424000689/type/journal_article |
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