On Induced Ramsey Numbers for Graphs with Bounded Maximum Degree

For graphsGandHwe writeG→indHif every 2-edge colouring ofGyields an induced monochromatic copy ofH. The induced Ramsey number forHis defined asrind(H)=min{|V(G)|: G→indH}. We show that for everyd⩾1 there exists an absolute constantcdsuch thatrind(Hn, d)⩽ncdfor every graphHn, dwithnvertices and the maximum degree at mostd. This confirms a conjecture suggested by W. T. Trotter.