Induced subgraphs of graphs with large chromatic number IX: rainbow paths

<p>We prove that for all integers <b><i>κ</i></b></p>,<b><i>s</i></b>≥<b>0</b> there exists <b><i>c</i></b> with the following property. Let <b><i>G</i></b> be a graph with clique number at most <b><i>κ</i></b> and chromatic number more than <b><i>c</i></b>. Then for every vertex-colouring (not necessarily optimal) of <b><i>G</i></b>, some induced subgraph of <b><i>G</i></b> is an <b><i>s</i></b>-vertex path, and all its vertices have different colours. This extends a recent result of Gyárfás and Sárközy (2016) who proved the same for graphs <b><i>G</i></b> with <b><i>κ</i></b>=<b>2</b> and girth at least five.