Arithmetic progressions of cycles in outer-planar graphs (Q1598841)

Let \(d\geq 3\) be an integer. In connection with a 1996 problem of Paul Erdős, the author proves that there exists a constant \(c>0\) such that if \(n\) is large enough, every outer-planar graph of order \(n\), in which every internal face has size at most \(d\) and the outer face is a cycle, contains a sequence of \(\exp\{c(\log n)^{1/3}-\log\log n\}\) cycles whose lengths form an arithmetic progression.