Universally Sparse Hypergraphs with Applications to Coding Theory

For fixed integers r ≥ 2, e ≥ 2, v ≥ r + 1, an r-uniform hypergraph is called ${{\mathcal{G}}_r}\left({v,e}\right)$-free if the union of any e distinct edges contains at least v+1 vertices. Let fr(n,v,e) denote the maximum number of edges in a ${{\mathcal{G}}_r}\left({v,e}\right)$-free r-uniform hypergraph on n vertices. Brown, Erdős and Sós showed in 1973 that there exist constants c1,c2 depending only on r,e,v such that${c_1}{n^{\frac{{er - v}}{{e - 1}}}} \leq {f_r}\left({n,v,e}\right) \leq {c_2}{n^{\left\lceil {\frac{{er - v}}{{e - 1}}} \right\rceil 1}}.$ For e − 1|er − v, the lower bound matches the upper bound up to a constant factor; whereas for e – 1ł er − v, it is a notoriously hard problem to determine the correct exponent of n. Our main result is an improvement ${f_r}\left({n,v,e}\right) = \Omega \left({{n^{\frac{{er - v}}{{e - 1}}}}{{(\log n)}^{\frac{1}{{e - 1}}}}}\right)$ for any r,e,v satisfying gcd(e − 1,er − v) = 1. Moreover, the hypergraph we constructed is not only ${{\mathcal{G}}_r}\left({v,e}\right)$-free but also universally ${{\mathcal{G}}_r}\left({ir - \left\lceil {\frac{{(i - 1)(er - v)}}{{e - 1}}} \right\rceil ,i}\right)$ -free for every 2 ≤ i ≤ e. Interestingly, our new lower bound provides improved constructions for several seemingly unrelated topics in Coding Theory, namely, Parent-Identifying Set Systems, uniform Combinatorial Batch Codes and optimal Locally Recoverable Codes.For a full version [1], see: https://arxiv.org/abs/1902.05903