Intersecting families of multisubsets with rank \(k\) (Q1352891)

Let \(E(s,n)=\{a=(a_1,\dots,a_n)\mid a_i\in\{0,\dots,s\}\) for each \(i\}\). A subset \(F\) of \(E(s,n)\) is called (statically) intersecting if for all \(a, b\in F\) there is some \(i\) such that \(a_i,b_i>0\), and \(F\) is said to be \(k\)-uniform if \(\sum^n_{i=1} a_i=k\) for all \(a\in F\). The author determines the maximum size of \(k\)-uniform intersecting families in \(E(s,n)\) for \(k\geq n\). The case \(k<n\) remains open. This case seems to be more difficult since at \(k=\lfloor \lambda^*_1n\rfloor\) there is some threshold and \(\lambda^*_1\) is a real number less than 1 defined in \textit{K. Engel} and \textit{P. Frankl} [An Erdös-Ko-Rado theorem for integer sequences of given rank, Eur. J. Comb. 7, 215-220 (1986; Zbl 0634.05004)].