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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.19327 |
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| _version_ | 1866912852087930880 |
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| author | Ho, Boon Suan |
| author_facet | Ho, Boon Suan |
| contents | In recent progress on the union-closed sets conjecture, a key lemma has been Boppana's entropy inequality: $h(x^2)\geϕxh(x)$, where $ϕ=(1+\sqrt5)/2$ and $h(x)=-x\log x-(1-x)\log(1-x)$. In this note, we prove that the generalized inequality $α_kh(x^k)\ge x^{k-1}h(x)$, first conjectured by Yuster, holds for real $k>1$, where $α_k$ is the unique positive solution to $x(1+x)^{k-1}=1$. This implies an analogue of the union-closed sets conjecture for approximate $k$-union closed set systems. We also formalize our proof in Lean 4. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19327 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A generalization of Boppana's entropy inequality Ho, Boon Suan Combinatorics Information Theory In recent progress on the union-closed sets conjecture, a key lemma has been Boppana's entropy inequality: $h(x^2)\geϕxh(x)$, where $ϕ=(1+\sqrt5)/2$ and $h(x)=-x\log x-(1-x)\log(1-x)$. In this note, we prove that the generalized inequality $α_kh(x^k)\ge x^{k-1}h(x)$, first conjectured by Yuster, holds for real $k>1$, where $α_k$ is the unique positive solution to $x(1+x)^{k-1}=1$. This implies an analogue of the union-closed sets conjecture for approximate $k$-union closed set systems. We also formalize our proof in Lean 4. |
| title | A generalization of Boppana's entropy inequality |
| topic | Combinatorics Information Theory |
| url | https://arxiv.org/abs/2601.19327 |