Supersaturation of odd linear cycles
Lirong Deng, Jie Han, Jiaxi Nie, Sam Spiro
TL;DR
This work advances the supersaturation theory for odd linear cycles in $r$-uniform hypergraphs, establishing a sharp conditional lower bound and a new upper bound for copies of $C^r_{2\ell+1}$ when the edge count is near the Tur\'an threshold. The authors develop a framework combining blow-ups, tensor products, and Sidorenko-style homomorphism densities to relate copies in expanded hypergraphs to those in the base structure, and employ a codegree-dichotomy to prove a main upper bound via induction on the uniformity $r$. A key conditional result shows that if dense high-girth hypergraphs exist with $n^{1+1/\ell-o(1)}$ edges, then the Sidorenko gap $s(C^r_{2\ell+1})$ must satisfy $\ge 1/((r-1)\ell-1)$, implying a concrete upper bound on the supersaturation exponent as a function of $(r,\ell)$. The paper also provides an unconditional weaker bound and situates these findings within the broader Sidorenko and Tur\'an frameworks, suggesting a likely precise value $s(C^r_{2\ell+1})=1/((r-1)\ell-1)$ for $r\ge3$, $\ell\ge2$ that awaits a full proof.
Abstract
An $r$-uniform linear cycle of length $\ell$, denoted by $C^r_{\ell}$, is an $r$-graph with $\ell$ edges $e_1,e_2,\dots,e_{\ell}$ where $e_i=\{v_{(r-1)(i-1)},v_{(r-1)(i-1)+1},\dots,v_{(r-1)i}\}$ (here $v_0=v_{(r-1)\ell}$). For $0<δ<1$ and $n$ sufficiently large, we show that every $n$-vertex $r$-graph $G$ with $n^{r-δ}$ edges contains at least $n^{(r-1)(2\ell+1)-δ(2\ell+1+\frac{4\ell-1}{(r-1)(2\ell+1)-3})-o(1)}$ copies of $C^r_{2\ell+1}$. Further, conditioning on the existence of dense high-girth hypergraphs, we show that there exists $n$-vertex $r$-graphs with $n^{r-δ}$ edges and at most $n^{(r-1)(2\ell+1)-δ(2\ell+1+\frac{1}{(r-1)\ell-1})+o(1)}$ copies of $C^r_{2\ell+1}$.