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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2410.07098 |
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| _version_ | 1866914968541069312 |
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| author | Bradač, Domagoj Liu, Hong Wu, Zhuo Xu, Zixiang |
| author_facet | Bradač, Domagoj Liu, Hong Wu, Zhuo Xu, Zixiang |
| contents | A well-known theorem of Nikiforov asserts that any graph with a positive $K_{r}$-density contains a logarithmic blowup of $K_r$. In this paper, we explore variants of Nikiforov's result in the following form. Given $r,t\in\mathbb{N}$, when a positive $K_{r}$-density implies the existence of a significantly larger (with almost linear size) blowup of $K_t$? Our results include:
For an $n$-vertex ordered graph $G$ with no induced monotone path $P_{6}$, if its complement $\overline{G}$ has positive triangle density, then $\overline{G}$ contains a biclique of size $Ω(\frac{n}{\log{n}})$. This strengthens a recent result of Pach and Tomon. For general $k$, let $g(k)$ be the minimum $r\in \mathbb{N}$ such that for any $n$-vertex ordered graph $G$ with no induced monotone $P_{2k}$, if $\overline{G}$ has positive $K_r$-density, then $\overline{G}$ contains a biclique of size $Ω(\frac{n}{\log{n}})$. Using concentration of measure and the isodiametric inequality on high dimensional spheres, we provide constructions showing that, surprisingly, $g(k)$ grows quadratically. On the other hand, we relate the problem of upper bounding $g(k)$ to a certain Ramsey problem and determine $g(k)$ up to a factor of 2.
Any incomparability graph with positive $K_{r}$-density contains a blowup of $K_r$ of size $Ω(\frac{n}{\log{n}}).$ This confirms a conjecture of Tomon in a stronger form. In doing so, we obtain a strong regularity type lemma for incomparability graphs with no large blowups of a clique, which is of independent interest.
We also prove that any $r$-comparability graph with positive $K_{(2h-2)^{r}+1}$-density contains a blowup of $K_h$ of size $Ω(n)$, where the constant $(2h-2)^{r}+1$ is optimal. The $\frac{n}{\log n}$ size of the blowups in all our results are optimal up to a constant factor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07098 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Clique density vs blowups Bradač, Domagoj Liu, Hong Wu, Zhuo Xu, Zixiang Combinatorics A well-known theorem of Nikiforov asserts that any graph with a positive $K_{r}$-density contains a logarithmic blowup of $K_r$. In this paper, we explore variants of Nikiforov's result in the following form. Given $r,t\in\mathbb{N}$, when a positive $K_{r}$-density implies the existence of a significantly larger (with almost linear size) blowup of $K_t$? Our results include: For an $n$-vertex ordered graph $G$ with no induced monotone path $P_{6}$, if its complement $\overline{G}$ has positive triangle density, then $\overline{G}$ contains a biclique of size $Ω(\frac{n}{\log{n}})$. This strengthens a recent result of Pach and Tomon. For general $k$, let $g(k)$ be the minimum $r\in \mathbb{N}$ such that for any $n$-vertex ordered graph $G$ with no induced monotone $P_{2k}$, if $\overline{G}$ has positive $K_r$-density, then $\overline{G}$ contains a biclique of size $Ω(\frac{n}{\log{n}})$. Using concentration of measure and the isodiametric inequality on high dimensional spheres, we provide constructions showing that, surprisingly, $g(k)$ grows quadratically. On the other hand, we relate the problem of upper bounding $g(k)$ to a certain Ramsey problem and determine $g(k)$ up to a factor of 2. Any incomparability graph with positive $K_{r}$-density contains a blowup of $K_r$ of size $Ω(\frac{n}{\log{n}}).$ This confirms a conjecture of Tomon in a stronger form. In doing so, we obtain a strong regularity type lemma for incomparability graphs with no large blowups of a clique, which is of independent interest. We also prove that any $r$-comparability graph with positive $K_{(2h-2)^{r}+1}$-density contains a blowup of $K_h$ of size $Ω(n)$, where the constant $(2h-2)^{r}+1$ is optimal. The $\frac{n}{\log n}$ size of the blowups in all our results are optimal up to a constant factor. |
| title | Clique density vs blowups |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.07098 |