On Sidorenko exponents of hypergraphs
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918152088059904 |
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| author | Lee, Hyunwoo |
| author_facet | Lee, Hyunwoo |
| contents | For an $r$-graph $F$, define Sidorenko exponent $s(F)$ as $$s(F):= \sup \{s \geq 0: \exists \text{$r$-graph $H$ s.t. } t_F(H) = t_{K^{(r)}_r} (H)^s > 0\},$$ where $t_{H_1}(H_2)$ denotes the homomorphism density of $H_1$ in $H_2$. The celebrated Sidorenko's conjecture states that $s(F) = e(F)$ holds for every bipartite graph $F$. It is known that for all $r \geq 3$, the $r$-uniform version of Sidorenko's conjecture is false, and only a few hypergraphs are known to be Sidorenko. In this paper, we discover a new broad class of Sidorenko hypergraphs and obtain general upper bounds on $s(F)$ for certain hypergraphs related to dominating hypergraphs. This makes progress toward a problem raised by Nie and Spiro. We also discover a new connection between Sidorenko exponents and upper bounds on the extremal numbers of a large class of hypergraphs, which generalizes the hypergraph analogue of Kővári--Sós--Turán theorem proved by Erdős. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08680 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Sidorenko exponents of hypergraphs Lee, Hyunwoo Combinatorics For an $r$-graph $F$, define Sidorenko exponent $s(F)$ as $$s(F):= \sup \{s \geq 0: \exists \text{$r$-graph $H$ s.t. } t_F(H) = t_{K^{(r)}_r} (H)^s > 0\},$$ where $t_{H_1}(H_2)$ denotes the homomorphism density of $H_1$ in $H_2$. The celebrated Sidorenko's conjecture states that $s(F) = e(F)$ holds for every bipartite graph $F$. It is known that for all $r \geq 3$, the $r$-uniform version of Sidorenko's conjecture is false, and only a few hypergraphs are known to be Sidorenko. In this paper, we discover a new broad class of Sidorenko hypergraphs and obtain general upper bounds on $s(F)$ for certain hypergraphs related to dominating hypergraphs. This makes progress toward a problem raised by Nie and Spiro. We also discover a new connection between Sidorenko exponents and upper bounds on the extremal numbers of a large class of hypergraphs, which generalizes the hypergraph analogue of Kővári--Sós--Turán theorem proved by Erdős. |
| title | On Sidorenko exponents of hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.08680 |