Rainbow Erdős-Sós Conjectures
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913673784590336 |
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| author | Crawford, Nicholas King, Dylan Spiro, Sam |
| author_facet | Crawford, Nicholas King, Dylan Spiro, Sam |
| contents | An edge colored graph is said to contain rainbow-$F$ if $F$ is a subgraph and every edge receives a different color. In 2007, Keevash, Mubayi, Sudakov, and Verstraëte introduced the \emph{rainbow extremal number} $\mathrm{ex}^*(n,F)$, a variant on the classical Turán problem, asking for the maximum number of edges in a $n$-vertex properly edge-colored graph which does not contain a rainbow-$F$. In the following years many authors have studied the asymptotic behavior of $\mathrm{ex}^*(n,F)$ when $F$ is bipartite. In the particular case that $F$ is a tree $T$, the infamous Erdös-Sós conjecture says that the extremal number of $T$ depends only on the size of $T$ and not its structure. After observing that such a pattern cannot hold for $\mathrm{ex}^*$ in the usual setting, we propose that the relative rainbow extremal number $\mathrm{ex}^*(Q_n,T)$ in the $n$-dimensional hypercube $Q_n$ will satisfy an Erdös-Sós type Conjecture and verify it for some infinite families of trees $T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00135 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rainbow Erdős-Sós Conjectures Crawford, Nicholas King, Dylan Spiro, Sam Combinatorics 05C15, 05C35 An edge colored graph is said to contain rainbow-$F$ if $F$ is a subgraph and every edge receives a different color. In 2007, Keevash, Mubayi, Sudakov, and Verstraëte introduced the \emph{rainbow extremal number} $\mathrm{ex}^*(n,F)$, a variant on the classical Turán problem, asking for the maximum number of edges in a $n$-vertex properly edge-colored graph which does not contain a rainbow-$F$. In the following years many authors have studied the asymptotic behavior of $\mathrm{ex}^*(n,F)$ when $F$ is bipartite. In the particular case that $F$ is a tree $T$, the infamous Erdös-Sós conjecture says that the extremal number of $T$ depends only on the size of $T$ and not its structure. After observing that such a pattern cannot hold for $\mathrm{ex}^*$ in the usual setting, we propose that the relative rainbow extremal number $\mathrm{ex}^*(Q_n,T)$ in the $n$-dimensional hypercube $Q_n$ will satisfy an Erdös-Sós type Conjecture and verify it for some infinite families of trees $T$. |
| title | Rainbow Erdős-Sós Conjectures |
| topic | Combinatorics 05C15, 05C35 |
| url | https://arxiv.org/abs/2502.00135 |