Multiply-intersecting families

Intersection problems occupy an important place in the theory of finite sets. One of the central notions is that of a r-wise r-intersecting family, that is, a collection holds for all choices of 1 < il < < i, < m. What is the maximal size m = m(n, r, t) of a r-wise t-intersecting family? Taking all subsets containing a fixed t-element set shows that m(n, r, 1) > 2 "-' holds for all n 3 f 2 0. One of the main results of the paper is that m(n,r,r)=2 "-' holds if and only if n n and we usually assume n 3 t, r 3 2. Even for t ,< n < t + r trivially m(n, r, t) = 2 "-' holds. For 0 < i 6 (n-t)/r define the families ~=~(n,r,t)=(Ac[n]:IAn[t+ri]l~t+(r-1)i). It is easy to see that 4 is r-wise c-intersecting, Id01 = 2 "-'. The basic open problem is the following. Conjecture 1.1 [F2]. m(n, r, t) = max{ IdI : 0 < i < (n-t)/r}.