Tur\'an numbers and batch codes

Combinatorial batch codes provide a tool for distributed data storage, with the feature of keeping privacy during information retrieval. Recently, Balachandran and Bhattacharya observed that the problem of constructing such uniform codes in an economic way can be formulated as a Tur\'an-type question on hypergraphs. Here we establish general lower and upper bounds for this extremal problem, and also for its generalization where the forbidden family consists of those $r$-uniform hypergraphs $H$ which satisfy the condition $k\ge |E(H)|> |V(H)|+q$ (for $k>q+r$ and $q> -r$ fixed). We also prove that, in the given range of parameters, the considered Tur\'an function is asymptotically equal to the one restricted to $|E(H)|=k$, studied by Brown, Erd\H{o}s and T. S\'os. Both families contain some $r$-partite members --- often called the `degenerate case', characterized by the equality $\lim_{n\to\infty} \ex(n,\cF)/n^r=0$ --- and therefore their exact order of growth is not known.