Andrásfai--Erdős--Sós theorem for the generalized triangle
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| Format: | Preprint |
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2024
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| _version_ | 1866916461723779072 |
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| author | Liu, Xizhi Ren, Sijie Wang, Jian |
| author_facet | Liu, Xizhi Ren, Sijie Wang, Jian |
| contents | The celebrated Andrásfai--Erdős--Sós Theorem from 1974 shows that every $n$-vertex triangle-free graph with minimum degree greater than $2n/5$ must be bipartite. Its extensions to $3$-uniform hypergraphs without the generalized triangle $F_5 = \{abc, abd, cde\}$ have been explored in several previous works such as~\cite{LMR23unif,HLZ24}, demonstrating the existence of $\varepsilon > 0$ such that for large $n$, every $n$-vertex $F_5$-free $3$-graph with minimum degree greater than $(1/9-\varepsilon) n^2$ must be $3$-partite.
We determine the optimal value for $\varepsilon$ by showing that for $n \ge 5000$, every $n$-vertex $F_5$-free $3$-graph with minimum degree greater than $4n^2/45$ must be $3$-partite, thus establishing the first tight Andrásfai--Erdős--Sós type theorem for hypergraphs. As a corollary, for all positive $n$, every $n$-vertex cancellative $3$-graph with minimum degree greater than $4n^2/45$ must be $3$-partite. This result is also optimal and considerably strengthens prior work, such as that by Bollobás~\cite{Bol74} and Keevash--Mubayi~\cite{KM04Cancel}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_20832 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Andrásfai--Erdős--Sós theorem for the generalized triangle Liu, Xizhi Ren, Sijie Wang, Jian Combinatorics The celebrated Andrásfai--Erdős--Sós Theorem from 1974 shows that every $n$-vertex triangle-free graph with minimum degree greater than $2n/5$ must be bipartite. Its extensions to $3$-uniform hypergraphs without the generalized triangle $F_5 = \{abc, abd, cde\}$ have been explored in several previous works such as~\cite{LMR23unif,HLZ24}, demonstrating the existence of $\varepsilon > 0$ such that for large $n$, every $n$-vertex $F_5$-free $3$-graph with minimum degree greater than $(1/9-\varepsilon) n^2$ must be $3$-partite. We determine the optimal value for $\varepsilon$ by showing that for $n \ge 5000$, every $n$-vertex $F_5$-free $3$-graph with minimum degree greater than $4n^2/45$ must be $3$-partite, thus establishing the first tight Andrásfai--Erdős--Sós type theorem for hypergraphs. As a corollary, for all positive $n$, every $n$-vertex cancellative $3$-graph with minimum degree greater than $4n^2/45$ must be $3$-partite. This result is also optimal and considerably strengthens prior work, such as that by Bollobás~\cite{Bol74} and Keevash--Mubayi~\cite{KM04Cancel}. |
| title | Andrásfai--Erdős--Sós theorem for the generalized triangle |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.20832 |