Erdős-Pósa from ball packing
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A classic theorem of Erdős and Pósa (1965) states that every graph has either k vertex-disjoint cycles or a set of O(k log k) vertices meeting all its cycles. We give a new proof of this theorem using a ball packing argument of Kühn and Osthus (2003). This approach seems particularly well suited for studying edge variants of the Erdős-Pósa theorem. As an illustration, we give a short proof of a theorem of Bruhn, Heinlein, and Joos (2019), that cycles of length at least ℓ have the so-called edge-Erdős-Pósa property. More precisely, we show that every graph G either contains k edge-disjoint cycles of length at least ℓ or an edge set F of size O(kℓ·log (kℓ)) such that G-F has no cycle of length at least ℓ. For fixed ℓ, this improves on the previously best known bound of O(k^2 log k +kℓ).
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