Hypergraphs of girth 5 and 6 and coding theory
Kathryn Haymaker, Michael Tait, Craig Timmons
TL;DR
This work studies Turán-type bounds for $r$-uniform hypergraphs avoiding Berge cycles of short length (girth $g=5,6$). It develops two complementary constructions that leverage coding theory (Dumer’s BCH codes and Sidon-coefficient frameworks) and number-theoretic Behrend-type sets to build large girth hypergraphs, yielding nontrivial lower bounds $\mathrm{ex}_r(N, \mathcal{C}_{<5})=\Omega_r\left(N^{\frac{3}{2}-o(1)}\right)$ for $r\in\{4,5,6\}$ and related bounds for larger $r$. The paper also connects these extremal constructions to coding-theory bounds, using recent sphere-packing improvements to sharpen the upper bounds for linear distance-$6$ codes, and discusses open obstacles to extending the approach to all $r$. It highlights the deep interplay between hypergraph Turán problems and coding theory, with implications for locally recoverable codes and related combinatorial structures. An open question (Behrend-over-field) invites further exploration of finite-field Behrend-type constructions for more general coefficient configurations.
Abstract
In this paper, we study the maximum number of edges in an $N$-vertex $r$-uniform hypergraph with girth $g$ where $g \in \{5,6 \}$. Writing $\textrm{ex}_r ( N, \mathcal{C}_{<g} )$ for this maximum, it is shown that $\textrm{ex}_r ( N , \mathcal{C}_{ < 5} ) = Ω_r ( N^{3/2 - o(1)} )$ for $r \in \{4,5,6 \}$. We address an unproved claim from [31] asserting a technique of Ruzsa can be used to show that this lower bound holds for all $r \geq 3$. We carefully explain one of the main obstacles that was overlooked at the time the claim from [31] was made, and show that this obstacle can be overcome when $r\in \{4,5,6\}$. We use constructions from coding theory to prove nontrivial lower bounds that hold for all $r \geq 3$. Finally, we use a recent result of Conlon, Fox, Sudakov, and Zhao to show that the sphere packing bound from coding theory may be improved when upper bounding the size of linear $q$-ary codes of distance $6$.