The extremal function for cycles of length ` mod k

Burr and Erdős conjectured that for each k, ` ∈ Z+ such that kZ + ` contains even integers, there exists a least function ck(`) such that any graph of average degree at least ck(`) contains a cycle of length ` mod k. This conjecture was proved by Bollobás, and many successive improvements of upper bounds on ck(`) appear in the literature. In this short note, for 1 6 ` 6 k, we show that ck(`) is proportional to the largest average degree of a C`-free graph on k vertices, which determines ck(`) up to an absolute constant. In particular, using known results on Turán numbers for even cycles, we obtain ck(`) = O(`k 2/`) for all even `, which is tight for ` ∈ {4, 6, 10}. Since the complete bipartite graph K`−1,n−`+1 has no cycle of length 2` mod k, it also shows ck(`) = Θ(`) for ` = Ω(log k).