A Refutation of the Pach-Tardos Conjecture for 0-1 Matrices
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911942416793600 |
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| author | Pettie, Seth Tardos, Gábor |
| author_facet | Pettie, Seth Tardos, Gábor |
| contents | The theory of forbidden 0-1 matrices generalizes Turan-style (bipartite) subgraph avoidance, Davenport-Schinzel theory, and Zarankiewicz-type problems, and has been influential in many areas, such as discrete and computational geometry, the analysis of self-adjusting data structures, and the development of the graph parameter twin width.
The foremost open problems in this area is to resolve the Pach-Tardos conjecture from 2005, which states that if a forbidden pattern $P\in\{0,1\}^{k\times l}$ is the bipartite incidence matrix of an acyclic graph (forest), then $\mathrm{Ex}(P,n) = O(n\log^{C_P} n)$, where $C_P$ is a constant depending only on $P$. This conjecture has been confirmed on many small patterns, specifically all $P$ with weight at most 5, and all but two with weight 6.
The main result of this paper is a clean refutation of the Pach-Tardos conjecture. Specifically, we prove that $\mathrm{Ex}(S_0,n),\mathrm{Ex}(S_1,n) \geq n2^{Ω(\sqrt{\log n})}$, where $S_0,S_1$ are the outstanding weight-6 patterns. We also prove sharp bounds on the entire class of alternating patterns $(P_t)$, specifically that for every $t\geq 2$, $\mathrm{Ex}(P_t,n)=Θ(n(\log n/\log\log n)^t)$. This is the first proof of an asymptotically sharp bound that is $ω(n\log n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02638 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Refutation of the Pach-Tardos Conjecture for 0-1 Matrices Pettie, Seth Tardos, Gábor Combinatorics Discrete Mathematics The theory of forbidden 0-1 matrices generalizes Turan-style (bipartite) subgraph avoidance, Davenport-Schinzel theory, and Zarankiewicz-type problems, and has been influential in many areas, such as discrete and computational geometry, the analysis of self-adjusting data structures, and the development of the graph parameter twin width. The foremost open problems in this area is to resolve the Pach-Tardos conjecture from 2005, which states that if a forbidden pattern $P\in\{0,1\}^{k\times l}$ is the bipartite incidence matrix of an acyclic graph (forest), then $\mathrm{Ex}(P,n) = O(n\log^{C_P} n)$, where $C_P$ is a constant depending only on $P$. This conjecture has been confirmed on many small patterns, specifically all $P$ with weight at most 5, and all but two with weight 6. The main result of this paper is a clean refutation of the Pach-Tardos conjecture. Specifically, we prove that $\mathrm{Ex}(S_0,n),\mathrm{Ex}(S_1,n) \geq n2^{Ω(\sqrt{\log n})}$, where $S_0,S_1$ are the outstanding weight-6 patterns. We also prove sharp bounds on the entire class of alternating patterns $(P_t)$, specifically that for every $t\geq 2$, $\mathrm{Ex}(P_t,n)=Θ(n(\log n/\log\log n)^t)$. This is the first proof of an asymptotically sharp bound that is $ω(n\log n)$. |
| title | A Refutation of the Pach-Tardos Conjecture for 0-1 Matrices |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2407.02638 |