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arXiv · 2212.13719

Tur\'an Numbers of Ordered Tight Hyperpaths

John P. Bright·Kevin G. Milans·Jackson Porter
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Abstract

An ordered hypergraph is a hypergraph $G$ whose vertex set $V(G)$ is linearly ordered. We find the Tur\'an numbers for the $r$-uniform $s$-vertex tight path $P^{(r)}_s$ (with vertices in the natural order) exactly when $r\le s < 2r$ and $n$ is even; our results imply $\mathrm{ex}_{>}(n,P^{(r)}_s)=(1-\frac{1}{2^{s-r}} + o(1))\binom{n}{r}$ when $r\le s<2r$. When $r\ge 2s$, the asymptotics of $\mathrm{ex}_{>}(n,P^{(r)}_s)$ remain open. For $r=3$, we give a construction of an $r$-uniform $n$-vertex hypergraph not containing $P^{(r)}_s$ which we conjecture to be asymptotically extremal.

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John P. Bright, Kevin G. Milans, Jackson Porter. 2022-12-28. Tur\'an Numbers of Ordered Tight Hyperpaths. https://arxiv.org/abs/2212.13719

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