On the Turán number of the hypercube
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910303316344832 |
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| author | Janzer, Oliver Sudakov, Benny |
| author_facet | Janzer, Oliver Sudakov, Benny |
| contents | In 1964, Erdős proposed the problem of estimating the Turán number of the $d$-dimensional hypercube $Q_d$. Since $Q_d$ is a bipartite graph with maximum degree $d$, it follows from results of Füredi and Alon, Krivelevich, Sudakov that $\mathrm{ex}(n,Q_d)=O_d(n^{2-1/d})$. A recent general result of Sudakov and Tomon implies the slightly stronger bound $\mathrm{ex}(n,Q_d)=o(n^{2-1/d})$. We obtain the first power-improvement for this old problem by showing that $\mathrm{ex}(n,Q_d)=O_d(n^{2-\frac{1}{d-1}+\frac{1}{(d-1)2^{d-1}}})$. This answers a question of Liu. Moreover, our techniques give a power improvement for a larger class of graphs than cubes.
We use a similar method to prove that any $n$-vertex, properly edge-coloured graph without a rainbow cycle has at most $O(n(\log n)^2)$ edges, improving the previous best bound of $n(\log n)^{2+o(1)}$ by Tomon. Furthermore, we show that any properly edge-coloured $n$-vertex graph with $ω(n\log n)$ edges contains a cycle which is almost rainbow: that is, almost all edges in it have a unique colour. This latter result is tight. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_02015 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Turán number of the hypercube Janzer, Oliver Sudakov, Benny Combinatorics In 1964, Erdős proposed the problem of estimating the Turán number of the $d$-dimensional hypercube $Q_d$. Since $Q_d$ is a bipartite graph with maximum degree $d$, it follows from results of Füredi and Alon, Krivelevich, Sudakov that $\mathrm{ex}(n,Q_d)=O_d(n^{2-1/d})$. A recent general result of Sudakov and Tomon implies the slightly stronger bound $\mathrm{ex}(n,Q_d)=o(n^{2-1/d})$. We obtain the first power-improvement for this old problem by showing that $\mathrm{ex}(n,Q_d)=O_d(n^{2-\frac{1}{d-1}+\frac{1}{(d-1)2^{d-1}}})$. This answers a question of Liu. Moreover, our techniques give a power improvement for a larger class of graphs than cubes. We use a similar method to prove that any $n$-vertex, properly edge-coloured graph without a rainbow cycle has at most $O(n(\log n)^2)$ edges, improving the previous best bound of $n(\log n)^{2+o(1)}$ by Tomon. Furthermore, we show that any properly edge-coloured $n$-vertex graph with $ω(n\log n)$ edges contains a cycle which is almost rainbow: that is, almost all edges in it have a unique colour. This latter result is tight. |
| title | On the Turán number of the hypercube |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2211.02015 |