Shift graphs and lower bounds on Ramsey numbers rk(l; r) Original Research Article

In this note we will obtain some lower bounds for the Ramsey numbers rk(l;r), where rk(l;r) is the least positive integer n such that for every coloring of the k-element subsets of an n-element set with r colors there always exists an l-element set, all of whose k-element subsets are colored the same. In particular, improving earlier results of Hirschfeld and complementing results of Erdös, Hajnal and Rado, we will show for r⩾ 3 that rk(l;r), l⩾k+1, grows like a tower, while determining the growth of rk(k+1;2) remains a problem.