Some multiple chromatic numbers of Kneser graphs

The Kneser graph K(m, n) (when m > 2n) has the n-subsets of an m-set as its vertices, two vertices being adjacent in K(m, n) whenever they are disjoint sets. The kth chromatic number of any graph G (denoted by ×k(G) ) is the least integer t such that the vertices can be assigned k-subsets of {1, 2, ..., t) with adjacent vertices receiving disjoint k-sets. S. Stahl has conjectured that, if k = qn - r where q ≥ 1 and 0 &le r &le n, then ×k-(K(m, n)) = qm - 2r. This expression is easily verified when r = 0; Stahl has also established its validity for q = 1, for m = 2n + 1 and for n = 2, 3. We show here that the expression is also valid for all q ≥ 2 in the following further classes of cases: (i) 2n + 1 &lem&len(2+r-1) (0&le r 1); (ii) 4 &le n &le 6 and 1 &le &le 2 (all m); (iii) 7 &le n &le 11 and r = 1 (all m); (iv) (n, r, m) = (7, 2, 18), (12, 1, 3 7), (12, 1, 38) or ( 13, 1, 40).