On an Extremal Hypergraph Problem Related to Combinatorial Batch Codes

Let $n, r, k$ be positive integers such that $3\leq k < n$ and $2\leq r \leq k-1$. Let $m(n, r, k)$ denote the maximum number of edges an $r$-uniform hypergraph on $n$ vertices can have under the condition that any collection of $i$ edges, span at least $i$ vertices for all $1 \leq i \leq k$. We are interested in the asymptotic nature of $m(n, r, k)$ for fixed $r$ and $k$ as $n \rightarrow \infty$. This problem is related to the forbidden hypergraph problem introduced by Brown, Erdős, and Sós and very recently discussed in the context of combinatorial batch codes. In this short paper we obtain the following results. {enumerate}[(i)] Using a result due to Erdős we are able to show $m(n, k, r) = o(n^r)$ for $7\leq k$, and $3 \leq r \leq k-1-\lceil\log k \rceil$. This result is best possible with respect to the upper bound on $r$ as we subsequently show through explicit construction that for $6 \leq k$, and $k-\lceil \log k \rceil \leq r \leq k-1, m(n, r, k) = Θ(n^r)$. This explicit construction improves on the non-constructive general lower bound obtained by Brown, Erdős, and Sós for the considered parameter values. For 2-uniform CBCs we obtain the following results. {enumerate} We provide exact value of $m(n, 2, 5)$ for $n \geq 5$. Using a result of Lazebnik,et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on $m(n, 2, k)$ ($Ω(n^{\frac{k+1}{k-1}})$) for all $k \geq 8$ and infinitely many values of $n$. We show $m(n, 2, k) = O(n^{1+\frac{1}{\lfloor\frac{k}{4}\rfloor}})$ by using a result due to Bondy and Simonovits, and also show $m(n, 2, k) = Θ(n^{3/2})$ for $k = 6, 7, 8$ by using a result of Kövari, Sós, and Turán.

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