Some intriguing upper bounds for separating hash families

An N × n matrix on q symbols is called {w1, . . . , wt}-separating if for arbitrary t pairwise disjoint column sets C1, . . . , Ct with |Ci| = wi for 1 ≤ i ≤ t, there exists a row f such that f(C1), . . . , f(Ct) are also pairwise disjoint, where f(Ci) denotes the values of Ci restricted to row f . Given integers N, q and w1, . . . , wt, denote by C(N, q, {w1, . . . , wt}) the maximal n such that a corresponding matrix does exist. The determination of C(N, q, {w1, . . . , wt}) has received remarkable attentions 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, {w1, . . . , wt}). 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, {w1, . . . , wt}), which significantly improve the previously known results.