Sparse hypergraphs with applications to coding theory

For fixed integers r \geq 3, e \geq 3, v \geq r + 1, an r-uniform hypergraph is called Gr(v, e)-free if the union of any e distinct edges contains at least v + 1 vertices. Brown, Erd\H os, and Sos showed that the maximum number of edges of such a hypergraph on n vertices, denoted as fr(n, v, e), satisfies \Omega (n ^{er e - -1 v} ) = fr(n, v, e) = O(n^{\lceil er e - -1 v \rceil} ). For sufficiently large n and e - 1 | er - v, the lower bound matches the upper bound up to a constant factor, which depends only on r, v, e; whereas for e - 1 \nmid er - v, in general it is a notoriously hard problem to determine the correct exponent of n. Among other results, we improve the above lower bound by showing that fr(n, v, e) = \Omega (n ^{er e - -1 v} (log n) e ^{-1 1} ) for any r, e, v satisfying gcd(e - 1, er - v) = 1. The hypergraph we constructed is in fact Gr(ir - \lceil ⁽i - 1)(er - v⁾ \rceil, i)-free for every 2 \leq i \leq e, and it has several interesting e - 1 applications in coding theory. The proof of the new lower bound is based on a novel application of the lower bound on the hypergraph independence number due to Duke, Lefmann, and Rodl.