Some intriguing upper bounds for separating hash families

An N - n matrix on q symbols is called {w ₁ ,..,w _t }-separating if for arbitrary t pairwise disjoint column sets C ₁ ,..,C _t with |C _i | = w _i for 1 ≤ i ≤ t, there exists a row f such that f(C ₁ ),.., f(C _t ) are also pairwise disjoint, where f(Ci) denotes the collection of components of C _i restricted to row f. Given integers N, q and w ₁ ,..,w _t , denote by C(N, q, {w ₁ ,..,w _t }) the maximal n such that a corresponding matrix does exist. The determination of C(N, q, {w ₁ ,..,w _t }) has received remarkable attention during the recent years. The main purpose of this paper is to introduce two novel methodologies to attack the upper bound of C(N, q, {w ₁ ,..,w _t }). The first one is a combination of the famous graph removal lemma in extremal graph theory and a Johnson-type recursive inequality in coding theory, and the second one is the probabilistic method. As a consequence, we obtain several intriguing upper bounds for some parameters of C(N, q, {w ₁ ,..,w _t }), which significantly improve the previously known results.