2020/05/06 by Maria Chudnovsky, Chudnovsky, Maria, Paul Seymour +1
Computer Science · Mathematics · Social Sciences · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Hungarian Social, Economic and Educational Studies #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2005.02896
openalex publication_date 2020/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A "hole-with-hat" in a graph G is an induced subgraph of G that consists of a cycle of length at least four, together with one further vertex that has exactly two neighbours in the cycle, adjacent to each other, and the "house" is the smallest, on five vertices. It is not known whether there exists ε>0 such that every graph G containing no house has a clique or stable set of cardinality at least |G|ε; this is one of the three smallest open cases of the Erdős-Hajnal conjecture and has been the subject of much study. We prove that there exists ε>0 such that every graph G with no hole-with-hat has a clique or stable set of cardinality at least |G|ε