Equivalence between Erdős–Hajnal and polynomial Rödl and Nikiforov conjectures
Matija Bucić, Jacob Fox, Huy Tuan Pham
Source abstract
Abstract It is well‐known that polynomial versions of theorems of Rödl and Nikiforov, as conjectured by Fox and Sudakov and Nguyen, Scott and Seymour imply the classical Erdős–Hajnal conjecture. In this note, we prove that these three conjectures are in fact equivalent, extending several previous particular results in this direction by Fox, Nguyen, Scott, and Seymour; Nguyen, Scott, and Seymour; and Gishboliner and Shapira. We deduce that the family of string graphs satisfies the polynomial Rödl conjecture. We also derive analogous results for hypergraphs, tournaments, ordered graphs, and colored graphs.
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