Absolutely avoidable order-size pairs in hypergraphs

For fixed integer $r\ge 2$, we call a pair $(m,f)$ of integers, $m\geq 1$, $0\leq f \leq \binom{m}{r}$, $absolutely$ $avoidable$ if there is $n_0$, such that for any pair of integers $(n,e)$ with $n>n_0$ and $0\leq e\leq \binom{n}{r}$ 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 $r\ge 3$ and $m \ge m_0$, either the pair $(m, \lfloor\binom mr/2\rfloor)$ or the pair $(m, \lfloor\binom{m}{r}/2\rfloor-m-1)$ is absolutely avoidable. Next, following the definition of Erdős, Füredi, Rothschild and Sós, we define the $density$ of a pair $(m,f)$ as $σ_r(m,f) = \limsup_{n \to \infty} \frac{|\{e : (n,e) \to (m,f)\}|}{\binom mr}$. We show that for $r\ge 3$ most pairs $(m,f)$ satisfy $σ_r(m,f)=0$, and that for $m > r$, there exists no pair $(m,f)$ of density 1.

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