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, Erd\H{o}s, and S\'os 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 Erd\H{o}s 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 $k-\lceil \log k \rceil \leq r \leq k-1, m(n, r, k) = \Theta(n^r)$. This explicit construction improves on the non-constructive general lower bound obtained by Brown, Erd\H{o}s, and S\'os 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 $m(n, 2, k)$ ($\Omega(n^{{\frac{k+1}{k-1}})$)} 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 $m(n, 2, k) = \Theta(n^{3/2})$ for $k = 6, 7, 8$ by using a result of K\"{o}vari, S\'os, and Tur\'{a}n.