Universally Sparse Hypergraphs with Applications to Coding Theory

For fixed integers r ≥ 2, e ≥ 2, v ≥ r + 1, an r-uniform hypergraph is called G r {v,e} -free if the union of any e distinct edges contains at least v+1 vertices. Let f_r(n,v,e) denote the maximum number of edges in a G {v,e} -free r-uniform hypergraph on n vertices. Brown, Erds and Sós showed in 1973 that there exist constants c₁,c₂ depending only on r,e,v such that c-1 {er - v} {e - 1} \leq {f-r} {n,v,e} \leq {c-2}{n lceil {{er - v}{e - 1} 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} {n,v,e} = Ω {er - v} {e - 1}( log n) {frac 1} {e - 1} for any r,e,v satisfying gcd(e - 1,er - v) = 1. Moreover, the hypergraph we constructed is not only G r {v,e}-free but also universally G r ir - lceil (i - 1)(er - v) {e - 1} rceil ,i -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