On edge-disjoint Ramsey numbers of stars

Abstract

For a graph \(F\) and a positive integer \(t\), the edge-disjoint Ramsey number \(ER_t(F)\) is the minimum positive integer \(n\) such that every red-blue coloring of the edges of the complete graph \(K_n\) of order \(n\) results in \(t\) pairwise edge-disjoint monochromatic copies of a subgraph isomorphic to \(F\). Since \(ER_1(F)\) is in fact the Ramsey number of \(F\), this concept extends the standard concept of Ramsey number. We investigate the edge-disjoint Ramsey numbers \(ER_t(K_{1, n})\) of the stars \(K_{1, n}\) of size \(n\). Formulas are established for \(ER_t(K_{1, n})\) for all positive integers \(n\) and \(t = 2, 3, 4\) and bounds are presented for \(ER_t(K_{1, n})\) for all positive integers \(n\) and \(t \ge 5\). Furthermore, exact values of \(ER_t(K_{1, n})\) are determined for \(n = 3, 4\) and several integers \(t \ge 5\).