Exact results for some extremal problems on expansions I
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| Format: | Preprint |
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2023
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| _version_ | 1866917818700660736 |
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| author | Liu, Xizhi Song, Jialei Yuan, Long-Tu |
| author_facet | Liu, Xizhi Song, Jialei Yuan, Long-Tu |
| contents | The expansion of a graph $F$, denoted by $F^3$, is the $3$-graph obtained from $F$ by adding a new vertex to each edge such that different edges receive different vertices. For large $n$, we establish tight upper bounds for:
The maximum number of edges in an $n$-vertex $3$-graph that does not contain $T^3$ for certain class $\mathcal{T}$ of trees, sharpening (partially) a result of Kostochka--Mubayi--Verstraëte.
The minimum number of colors needed to color the complete $n$-vertex $3$-graph to ensure the existence of a rainbow copy of $F^3$ when $F$ is a graph obtained from some tree $T\in \mathcal{T}$ by adding a new edge, extending anti-Ramsey results on $P_{2t}^3$ by Gu--Li--Shi and $C_{2t}^3$ by Tang--Li--Yan.
The maximum number of edges in an $n$-vertex $3$-graph whose shadow does not contain the shadow of $C_{k}^3$ or $T^3$ for $T\in \mathcal{T}$, answering a question of Lv \etal on generalized Turán problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_01736 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Exact results for some extremal problems on expansions I Liu, Xizhi Song, Jialei Yuan, Long-Tu Combinatorics The expansion of a graph $F$, denoted by $F^3$, is the $3$-graph obtained from $F$ by adding a new vertex to each edge such that different edges receive different vertices. For large $n$, we establish tight upper bounds for: The maximum number of edges in an $n$-vertex $3$-graph that does not contain $T^3$ for certain class $\mathcal{T}$ of trees, sharpening (partially) a result of Kostochka--Mubayi--Verstraëte. The minimum number of colors needed to color the complete $n$-vertex $3$-graph to ensure the existence of a rainbow copy of $F^3$ when $F$ is a graph obtained from some tree $T\in \mathcal{T}$ by adding a new edge, extending anti-Ramsey results on $P_{2t}^3$ by Gu--Li--Shi and $C_{2t}^3$ by Tang--Li--Yan. The maximum number of edges in an $n$-vertex $3$-graph whose shadow does not contain the shadow of $C_{k}^3$ or $T^3$ for $T\in \mathcal{T}$, answering a question of Lv \etal on generalized Turán problems. |
| title | Exact results for some extremal problems on expansions I |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2310.01736 |