Erdős Problems

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OPEN

Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1<a_2<\cdots $ be the lengths of cycles in $G$. Is it true that \[\sum\frac{1}{a_i}\gg \log k?\] Is the sum $\sum\frac{1}{a_i}$ minimised when $G$ is a complete bipartite graph?

#65: [Er74d][Er75][Er81][Er95]

graph theory, cycles

A problem of Erdős and Hajnal. Gyárfás, Komlós, and Szemerédi [GyKoSz84] have proved that this sum is $\gg \log k$. Liu and Montgomery [LiMo20] have proved the asymptotically sharp lower bound of $\geq (\tfrac{1}{2}-o(1))\log k$.