Prime-length cycles and all zero lower-density length sets lack the 1/t-integral Erdős-Pósa property even in planar graphs, with a further failure for porous sets in projective planar graphs.
Pascal Gollin, Kevin Hendrey, O-joung Kwon, Sang-il Oum, and Youngho Yoo
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The Erd\H{o}s-P\'osa property for prime-length cycles fails (and beyond)
Prime-length cycles and all zero lower-density length sets lack the 1/t-integral Erdős-Pósa property even in planar graphs, with a further failure for porous sets in projective planar graphs.