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- W4289782012 abstract "For fixed integer $rge 2$, we call a pair $(m,f)$ of integers, $mgeq 1$, $0leq f leq binom{m}{r}$, $absolutely$ $avoidable$ if there is $n_0$, such that for any pair of integers $(n,e)$ with $n>n_0$ and $0leq eleq binom{n}{r}$ there is an $r$-uniform hypergraph on $n$ vertices and $e$ edges that contains no induced sub-hypergraph on $m$ vertices and $f$ edges. Some pairs are clearly not absolutely avoidable, for example $(m,0)$ is not absolutely avoidable since any sufficiently sparse hypergraph on at least $m$ vertices contains independent sets on $m$ vertices. Here we show that for any $rge 3$ and $m ge m_0$, either the pair $(m, lfloorbinom mr/2rfloor)$ or the pair $(m, lfloorbinom{m}{r}/2rfloor-m-1)$ is absolutely avoidable. Next, following the definition of ErdH{o}s, Furedi, Rothschild and S'os, we define the $density$ of a pair $(m,f)$ as $sigma_r(m,f) = limsup_{n to infty} frac{|{e : (n,e) to (m,f)}|}{binom mr}$. We show that for $ rge 3$ most pairs $(m,f)$ satisfy $sigma_r(m,f)=0$, and that for $m > r$, there exists no pair $(m,f)$ of density 1." @default.
- W4289782012 created "2022-08-04" @default.
- W4289782012 creator A5040212186 @default.
- W4289782012 date "2022-05-30" @default.
- W4289782012 modified "2023-10-16" @default.
- W4289782012 title "Absolutely avoidable order-size pairs in hypergraphs" @default.
- W4289782012 doi "https://doi.org/10.48550/arxiv.2205.15197" @default.
- W4289782012 hasPublicationYear "2022" @default.
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