Small transversals in hypergraphs

For each positive integer k, we consider the set A k of all ordered pairs [a, b] such that in every k-graph with n vertices and m edges some set of at most am + bn vertices meets all the edges. We show that each A k with k > 2 has infinitely many extreme points and conjecture that, for every positive e, it has only finitely many extreme points [a, b] with a _> e. With the extreme points ordered by the first coordinate, we identify the last two extreme points of every Ak, identify the last three extreme points of A3, and describe A 2 completely. A by-product of our arguments is a new algorithmic proof of Tur~n's theorem. 1. The problem A k-graph is an ordered pair (V, E) such that V is a finite set and E is a set of distinct k-point subsets of V. The elements of V are the vertices of the k-graph and the elements of E are the edges of the k-graph. We reserve the letters n and m for the number of vertices and for the number of edges, respectively, of a k-graph H, and similarly for ni, mi, H/. A transversal (or a cover or a blocking set) in a k-graph is a set of vertices that meets all the edges; we let "r(H) denote the smallest size of a transversal in a k-graph H. A problem of Turs [6] can be stated as the problem of determining the smallest t(n,m, k) such that every k-graph H with n vertices and m edges has T(H) <_ t(n, m, k). Trivially, t(n, m, 1) = m. Turin [5] evaluated t(n, m, 2); the case of k _> 3 remains unsolved. This is hardly surprising as Turs problem subsumes other notoriously difficult combinatorial problems: for instance, t(lll, 111,100) _> 3 if and only if a projective plane of order 10 exists. (To see this, consider any 100-graph H such that H-(V, E) with IV] = IEI = 111; define H* to be the ll-graph (V, E*) such that A 9 E* if and only if V-A 9 E. Now T(H) ~ 3 if and only if every two points of V lie in a common edge of H*, which is the case if and only if H* is a projective plane of order 10.) We propose an easier variation on Turs …