A solution to Erdős and Hajnal’s odd cycle problem

In 1981, Erdős and Hajnal asked whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. Let C ( G ) \mathcal {C}(G) be the set of cycle lengths in a graph G G and let C o d d ( G ) \mathcal {C}_{\mathrm {odd}}(G) be the set of odd numbers in C ( G ) \mathcal {C}(G) . We prove that, if G G has chromatic number k k , then ∑ ℓ ∈ C o d d ( G ) 1 / ℓ ≥ ( 1 / 2 − o k ( 1 ) ) log ⁡ k \sum _{\ell \in \mathcal {C}_{\mathrm {odd}}(G)}1/\ell \geq (1/2-o_k(1))\log k . This solves Erdős and Hajnal’s odd cycle problem, and, furthermore, this bound is asymptotically optimal. In 1984, Erdős asked whether there is some d d such that each graph with chromatic number at least d d (or perhaps even only average degree at least d d ) has a cycle whose length is a power of 2. We show that an average degree condition is sufficient for this problem, solving it with methods that apply to a wide range of sequences in addition to the powers of 2. Finally, we use our methods to show that, for every k k , there is some d d so that every graph with average degree at least d d has a subdivision of the complete graph K k K_k in which each edge is subdivided the same number of times. This confirms a conjecture of Thomassen from 1984.