Universally Sparse Hypergraphs with Applications to Coding Theory
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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_r(v,e)-free r-uniform hypergraph on n vertices. Brown, Erdős and Sós showed in 1973 that there exist constants c_1,c_2 depending only on r,e,v such that c_1n^er-v/e-1< f_r(n,v,e)< c_2n^er-v/e-1. For e-1|er-v, the lower bound matches the upper bound up to a constant factor; whereas for e-1∤ er-v, it is a notoriously hard problem to determine the correct exponent of n for general r,e,v. Our main result is an improvement on the above lower bound by a ( n)^1/e-1 factor f_r(n,v,e)=Ω(n^er-v/e-1( n)^1/e-1) for any r,e,v satisfying (e-1,er-v)=1. Moreover, the hypergraph we constructed is not only G_r(v,e)-free but also universally G_r(ir-(i-1)(er-v)/e-1,i)-free for every 1< i< e. Interestingly, our new lower bound provides improved constructions for several seemingly unrelated objects in Coding Theory, namely, Parent-Identifying Set Systems, uniform Combinatorial Batch Codes and optimal Locally Recoverable Codes. The proof of the main result is based on a novel application of the well-known lower bound on the hypergraph independence number due to Ajtai, Komlós, Pintz, Spencer, and Szemerédi, and Duke, Lefmann, and Rödl.
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