ON THE MATCHING NUMBER AND THE INDEPENDENCE NUMBER OF A RANDOM INDUCED SUBHYPERGRAPH OF A HYPERGRAPH

For r >= 2, let H be an r-uniform hypergraph with n vertices and m hyperedges. Let R be a random vertex set obtained by choosing each vertex of H independently with probability p. Let H[R] be the subhypergraph of H induced on R. We obtain an upper bound on the matching number nu(H[R]) and a lower bound on the independence number alpha(H[R]) of H[R]. First, we show that if mp(r) >= log n, then nu(H[R]) <= 2e(l)mp(r) with probability at least 1 - 1/n(l) for each positive integer l. It is best possible up to a constant factor depending only on l if m <= n/r. Next, we show that if mp(r) >= log n, then alpha(H[R]) >= np-root 3lnp log n - 2re(l)mp(r) with probability at least 1 - 3/n(l).