The Erdős-Pósa property for prime-length cycles fails (and beyond)
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arXiv
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| Format: | Preprint |
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2026
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| author | Gorsky, Maximilian Hendrey, Kevin Huynh, Tony |
| author_facet | Gorsky, Maximilian Hendrey, Kevin Huynh, Tony |
| contents | We prove that for every $t \in \mathbb{N}$, prime-length cycles do not have the $\frac{1}{t}$-integral Erdős-Pósa property, even when restricted to planar graphs. We in fact prove a more general density result. For every $t \in \mathbb{N}$ and every subset $L \subseteq \mathbb{N}$ with lower density zero, the set of cycles whose length is in $L$ do not have the $\frac{1}{t}$-integral Erdős-Pósa property, even when restricted to planar graphs. We also consider a less restrictive density condition on $L$, called porous, where the complement of $L$ contains arbitrarily long sequences of consecutive integers. We prove that for every porous set $L \subseteq \mathbb{N}$, the set of cycles whose length is in $L$ do not have the Erdős-Pósa property, even when restricted to projective planar graphs. Our results partially answer a question of Gollin, Hendrey, Kwon, Oum, and Yoo [Math. Ann., 393(2):2507-2559, 2025]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_04938 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Erdős-Pósa property for prime-length cycles fails (and beyond) Gorsky, Maximilian Hendrey, Kevin Huynh, Tony Combinatorics Number Theory 05B40, 05C38, 11A41 We prove that for every $t \in \mathbb{N}$, prime-length cycles do not have the $\frac{1}{t}$-integral Erdős-Pósa property, even when restricted to planar graphs. We in fact prove a more general density result. For every $t \in \mathbb{N}$ and every subset $L \subseteq \mathbb{N}$ with lower density zero, the set of cycles whose length is in $L$ do not have the $\frac{1}{t}$-integral Erdős-Pósa property, even when restricted to planar graphs. We also consider a less restrictive density condition on $L$, called porous, where the complement of $L$ contains arbitrarily long sequences of consecutive integers. We prove that for every porous set $L \subseteq \mathbb{N}$, the set of cycles whose length is in $L$ do not have the Erdős-Pósa property, even when restricted to projective planar graphs. Our results partially answer a question of Gollin, Hendrey, Kwon, Oum, and Yoo [Math. Ann., 393(2):2507-2559, 2025]. |
| title | The Erdős-Pósa property for prime-length cycles fails (and beyond) |
| topic | Combinatorics Number Theory 05B40, 05C38, 11A41 |
| url | https://arxiv.org/abs/2605.04938 |