Forbidden subgraphs in divisor graphs and an Erdős divisibility problem
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914489023070208 |
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| author | Davis, Damek |
| author_facet | Davis, Damek |
| contents | Erdős asked for the largest size $f(n)$ of a subset of $\{1,\dots,n\}$ with no element dividing two others. We show that $f(n)=c_2\,n+o(n)$ for an effectively computable constant $c_2$, and moreover that the number $q(n)$ of such subsets satisfies $q(n)=β_2^{n+o(n)}$ for a computable constant $β_2$. To prove this, we recast the divisibility constraint as forbidding a certain directed subgraph in the divisor graph on $\{1,\dots,n\}$ and prove a more general result: for any finite family of connected forbidden subgraphs of the divisor graph, both the extremal density and counting rate are effectively computable. The proof uses a theorem of McNew on local statistics of divisor graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17613 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Forbidden subgraphs in divisor graphs and an Erdős divisibility problem Davis, Damek Combinatorics Number Theory Erdős asked for the largest size $f(n)$ of a subset of $\{1,\dots,n\}$ with no element dividing two others. We show that $f(n)=c_2\,n+o(n)$ for an effectively computable constant $c_2$, and moreover that the number $q(n)$ of such subsets satisfies $q(n)=β_2^{n+o(n)}$ for a computable constant $β_2$. To prove this, we recast the divisibility constraint as forbidding a certain directed subgraph in the divisor graph on $\{1,\dots,n\}$ and prove a more general result: for any finite family of connected forbidden subgraphs of the divisor graph, both the extremal density and counting rate are effectively computable. The proof uses a theorem of McNew on local statistics of divisor graphs. |
| title | Forbidden subgraphs in divisor graphs and an Erdős divisibility problem |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2604.17613 |