  The well know theorem of Tverberg states that if n > (d+1)(r-1) then one can
partition any set of n points in R^d to r disjoint subsets whose convex hulls
have a common point. The numbers T(d,r) = (d + 1)(r - 1) + 1 are known as
Tverberg numbers. Reay asks the following question: if we add an additional
parameter k (1 < k < r+1) what is the minimal number of points we need in order
to guarantee that there exists an r partition of them such that any k of the r
convex hulls intersect. This minimal number is denoted by T(d,r,k). Reay
conjectured that T(d,r,k) = T(d,r) for all d,r and k. In this article we prove
that this is true for the following cases: when k > [ (d+3)/2 ]-1 or when d <
rk/(r-k)-1 and for the specific values d = 3; r = 4; k = 2 and d = 5; r = 3; k
= 2.
