Connected Counterexamples for Target Ramsey Numbers

Abstract

We answer an open problem of Chartrand and Zhang concerning target Ramsey numbers. We prove that there exist infinitely many connected graphs whose target Ramsey number is strictly larger than their ordinary Ramsey number. The proof combines an elementary counting lower bound with asymptotic Ramsey results for balanced grid graphs. As a complementary result, we present an independent asymptotic construction based on Ramsey numbers of multiple copies of a fixed graph.