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Uniform hypergraphs of girth $6$ and $8$ from generalized polygons

Nikolai Parvatov

TL;DR

The paper addresses maximizing edge counts in r-uniform hypergraphs with prescribed girth g ∈ {6,8} as the order N grows. It introduces a construction pipeline that uses generalized polygons (hexagons and octagons) to build base hypergraphs from bipartite, bi-regular graphs and then recurses via edge replacement to amplify edge counts while preserving girth. The authors establish new asymptotic lower bounds ex_r(N,6) ≥ N^{11/8 − c/√log_2 N} and ex_r(N,8) ≥ N^{11/9 − d/√log_2 N} for large N, thereby advancing the known landscape of extremal hypergraph girth problems. The work leverages finite geometry objects like generalized hexagons and Ree-Tits octagons to construct large, high-girth hypergraphs with tunable uniformity, offering a method with potential applicability to coding theory and combinatorial design.

Abstract

Let $ex_r(N,g)$ be the maximum number of edges in an $r$-uni\-form hypergraph on $N$ vertices with girth at least $g$. We are interested in the asymptotic behavior of this value when $N$ is increasing but parameters $g\in\{6,8\}$ and $r\geq3$ are fixed. It is shown that for some positive constants $c$ and $d$, any integer $r\geq3$ and all sufficiently large integers $N$ the inequalities $ex_r(N,6)\geq N^{\frac{11}{8}-\frac{c}{\sqrt{\log N}}}$ and $ex_r(N,8)\geq N^{\frac{11}{9}-\frac{d}{\sqrt{\log N}}}$ hold.

Uniform hypergraphs of girth $6$ and $8$ from generalized polygons

TL;DR

The paper addresses maximizing edge counts in r-uniform hypergraphs with prescribed girth g ∈ {6,8} as the order N grows. It introduces a construction pipeline that uses generalized polygons (hexagons and octagons) to build base hypergraphs from bipartite, bi-regular graphs and then recurses via edge replacement to amplify edge counts while preserving girth. The authors establish new asymptotic lower bounds ex_r(N,6) ≥ N^{11/8 − c/√log_2 N} and ex_r(N,8) ≥ N^{11/9 − d/√log_2 N} for large N, thereby advancing the known landscape of extremal hypergraph girth problems. The work leverages finite geometry objects like generalized hexagons and Ree-Tits octagons to construct large, high-girth hypergraphs with tunable uniformity, offering a method with potential applicability to coding theory and combinatorial design.

Abstract

Let be the maximum number of edges in an -uni\-form hypergraph on vertices with girth at least . We are interested in the asymptotic behavior of this value when is increasing but parameters and are fixed. It is shown that for some positive constants and , any integer and all sufficiently large integers the inequalities and hold.
Paper Structure (7 sections, 4 theorems, 31 equations)

This paper contains 7 sections, 4 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Two observations
  3. The construction
  4. Basic construction
  5. The generalized construction
  6. The lower bound for $\mathop{\mathrm{ex}}\nolimits_r(N,6)$
  7. Hypergaphs from generalized octagons

Key Result

Lemma 1

Suppose $p$ is a prime number and $m,n,r$ are positive integer numbers such that $p^{m-1}\geq5$ and $2\leq r\leq1+p^m$. There exists an $r$-uni-form hypergraph $\Gamma^r_{p,m,n}$ of girth at least $6$ with $v(Q_{p,m,n})$ vertices and at least $p^{\frac{11}{8}(9^n(m+\frac{1}{8})-(n+m+\frac{1}{8}))}$

Theorems & Definitions (4)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Theorem 2