On facets of stable set polytopes of claw-free graphs with stability number three

Providing a complete description of the stable set polytopes of claw-free graphs is a long-standing open problem since almost twenty years. Eisenbrandt et al. recently achieved a breakthrough for the subclass of quasi-line graphs. As a consequence, every non-trivial facet of their stable set polytope is of the form k∑v∈V1xv+(k+1)∑v∈V2xv⩽b for some positive integers k and b, and non-empty sets of vertices V1 and V2. Roughly speaking, this states that the facets of the stable set polytope of quasi-line graphs have at most two left coefficients. For stable set polytopes of claw-free graphs with maximum stable set size at least four, Stauffer conjectured in 2005 that this still holds. It is already known that some stable set polytopes of claw-free graphs with maximum stable set size three may have facets with up to 5 left coefficients.We prove that the situation is even worse: for every positive integer b, there is a claw-free graph with stability number three whose stable set polytope has a facet with b left coefficients.