On Sidorenko exponents of hypergraphs
Hyunwoo Lee
TL;DR
This work addresses the problem of determining Sidorenko exponents $s(F)$ for $r$-partite $r$-graphs by developing a general framework built on link profiles, dominating hypergraphs, and the tensor power trick. It proves new wide-ranging Sidorenko criteria (Theorem 1) and domination-based bounds (Theorem 2), and connects these exponents to extremal numbers through a KST-type bound (Theorem 3), enabling improved bounds for families such as 3-uniform tight cycles, sparse hypergraphs, and grids. The approach yields a cohesive pathway from structural hypergraph properties to quantitative Sidorenko-type inequalities and extremal consequences, with concrete applications to $s(C^{(3)}_{3\ell})$, as well as guiding conjectures for higher uniformities. Overall, the paper advances the understanding of how domination and norming phenomena govern Sidorenko-type inequalities in hypergraphs and their extremal implications, offering a versatile toolkit for bounding $s(F)$ across broad hypergraph families.
Abstract
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.