Subdivisions of four blocks cycles in digraphs with large chromatic number

A cycle with four blocks C(k(1), k(2), k(3), k(4)) is an oriented cycle formed of four blocks of lengths k(1), k(2), k(3) and k(4) respectively. We conjecture that for every positive integers k(1), k(2), k(3), k(4), there is an integer g(k(1), k(2), k(3), k(4)) such that every strongly connected digraph with chromatic number greater than g(k(1), k(2), k(3), k(4)) contains a subdivision of C(k(1), k(2), k(3), k(4)). As evidence, we prove this conjecture for k(2) = k(3) = k(4) = 1. (C) 2021 Published by Elsevier B.V.