On a generalization of Ramsey numbers (Q2556333)

Define \(m=N(l_1,k_1;l_2,k_2;r)\) as the smallest integer with the property that if the \(r\)-tuples of a set of \(m\) elements are arbitrarily split into two classes then for \(i=1\) or \(2\) there exists a subset of size \(l_i\) each of whose subsets of size \(k_i\) lies in some \(r\)-subset of the \(i\)-th class. \(N(l_1,r;l_2;r;r)\) is the Ramsey number \(N(l_1,l_2;r)\). The authors prove that if \(k_1+k_2=r+1\) then \[ N(l_1,k_1;l_2,k_2;r)=l_1+l_2-k_1-k_2+1. \] If \(k+1+k_2=r+2\) the authors prove \[ 2^{c_1l}<N(l_1,k_1;l_2,k_2;r)<2^{c_2l}. \]