By a very simple argument, we prove that if l, m, n {0, 1, 2, . . . } then l k=0 (-1)m-k l k m - k n 2k k - 2l + m = l k=0 l k 2k n n - l m + n - 3k - l . On the basis of this ide...
For a given positive integer k, sr(m, k) denotes the minimal positive integer such that every coloring of [n], n ≥ sr(m, k), that uses each color at most k times, yields a rainb...
It is known that for every integer k ≥ 4, each k-map graph with n vertices has at most kn − 2k edges. Previously, it was open whether this bound is tight or not. We show that ...