Abstract. The numbers which are traditionally named in honor of Paul Turfin were introduced by him as a generalization of a problem he solved in 1941. The general problem of Tur~in having an extremely simple formulation but being extremely hard to solve, has become one of the most fascinating extremal problems in combinatorics. We describe the present situation and list conjectures which are not so hopeless.
The Definition and Equivalent FormulationsA system of r-element subsets (blocks) of an n-element set Xn is called a Turin (n, k, r)-system if every k-element subset of X, contains at least one of the blocks. The Turrn number T(n, k, r) is the minimum size of such a system. The problem of determining T(n, k, r) was posed by Paul Turin [57].The Tur~m numbers sometimes appear in different notation. For instance, the covering number C(n, m, p) is defined as the minimum number of m-element subsets of X, needed to cover all p-element subsets (n > m > p). Obviously,
C(n,m,p) = T(n,n -p,n -m).Let U(n, q, r) be the minimum number of subsets of Xn whose sizes are at least r and for which the size of any transversal (i.e. a subset intersecting each of them) is at least q. It is easy to see that U(n,q,r) = T(n,n -q + 1,r).(1) It was shown in I45] that T(n, k, r) is equal to the minimal length of a disjunctive normal form with n Boolean variables, which takes value 1 if at least k variables are equal to 1, and takes value 0 if less than r variables are equal to 1. Schrnheim [39], and independently, Katona, Nemetz and Simonovits [19] showed that I 1 r (n, k, r) > r(n -1, k, r) .
Recursive Inequalitiesn~r
