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Fractional Clique Decompositions of Dense Hypergraphs

Michelle Delcourt, Thomas Lesgourgues, Luke Postle

TL;DR

The paper advances the fractional relaxation of the hypergraph design decomposition problem by proving that the fractional $K_q^r$-decomposition threshold satisfies $\delta^*_{K_q^r}\le 1-\frac{C}{q^{r-1+o(1)}}$ for all $r\ge3$ and large $q$, for any fixed $\varepsilon>0$. The authors develop a hypergraph-appropriate adaptation of Montgomery's method, centered on three pillars: a direct inclusion-exclusion construction to obtain a fractional $K_q^r$-decomposition after deleting a single edge, a fixing lemma to upgrade almost decompositions to exact fractions, and a sampling-based strategy to generate almost decompositions while controlling the number of missing matchings. They further show that removing a bounded number of matchings from the complete hypergraph preserves a fractional $K_q^r$-decomposition, and they assemble these pieces via a concatenation framework to produce a full fractional decomposition for dense hypergraphs. Combined with Rödl–Schacht–Siggers–Tokushige, these results yield approximate $K_q^r$-decompositions for large hypergraphs, moving the fractional bound closer to the conjectured regime. The work provides new tools—such as a robust concatenation lemma and a sampling-lemma construction—that are likely to influence future advances on hypergraph design decompositions.

Abstract

In 2014, Keevash famously proved the existence of $(n,q,r)$-Steiner systems as part of settling the Existence Conjecture of Combinatorial Designs (dating from the mid-1800s). In 2020, Glock, Kühn, and Osthus conjectured a minimum degree generalization: specifically that minimum $(r-1)$-degree at least $(1-\frac{C}{q^{r-1}})n$ suffices to guarantee that every sufficiently large $K_q^r$-divisible $r$-uniform hypergraph on $n$ vertices admits a $K_q^r$-decomposition (where $C$ is a constant that is allowed to depend on $r$ but not $q$). The best-known progress on this conjecture is from the second proof of the Existence Conjecture by Glock, Kühn, Lo, and Osthus in 2016 who showed that $(1-\frac{C}{q^{2r}})n$ suffices. The fractional relaxation of the conjecture is crucial to improving the bound; for that, only the slightly better bound of $(1-\frac{C}{q^{2r-1}})n$ was known due to Barber, Kühn, Lo, Montgomery, and Osthus from 2017. Our main result is to prove that $(1-\frac{C}{q^{r-1+o(1)}})n$ suffices for the fractional relaxation. Combined with the work of R{ö}dl, Schacht, Siggers, and Tokushige from 2007, this also shows that such hypergraphs admit approximate $K_q^r$-decompositions.

Fractional Clique Decompositions of Dense Hypergraphs

TL;DR

The paper advances the fractional relaxation of the hypergraph design decomposition problem by proving that the fractional -decomposition threshold satisfies for all and large , for any fixed . The authors develop a hypergraph-appropriate adaptation of Montgomery's method, centered on three pillars: a direct inclusion-exclusion construction to obtain a fractional -decomposition after deleting a single edge, a fixing lemma to upgrade almost decompositions to exact fractions, and a sampling-based strategy to generate almost decompositions while controlling the number of missing matchings. They further show that removing a bounded number of matchings from the complete hypergraph preserves a fractional -decomposition, and they assemble these pieces via a concatenation framework to produce a full fractional decomposition for dense hypergraphs. Combined with Rödl–Schacht–Siggers–Tokushige, these results yield approximate -decompositions for large hypergraphs, moving the fractional bound closer to the conjectured regime. The work provides new tools—such as a robust concatenation lemma and a sampling-lemma construction—that are likely to influence future advances on hypergraph design decompositions.

Abstract

In 2014, Keevash famously proved the existence of -Steiner systems as part of settling the Existence Conjecture of Combinatorial Designs (dating from the mid-1800s). In 2020, Glock, Kühn, and Osthus conjectured a minimum degree generalization: specifically that minimum -degree at least suffices to guarantee that every sufficiently large -divisible -uniform hypergraph on vertices admits a -decomposition (where is a constant that is allowed to depend on but not ). The best-known progress on this conjecture is from the second proof of the Existence Conjecture by Glock, Kühn, Lo, and Osthus in 2016 who showed that suffices. The fractional relaxation of the conjecture is crucial to improving the bound; for that, only the slightly better bound of was known due to Barber, Kühn, Lo, Montgomery, and Osthus from 2017. Our main result is to prove that suffices for the fractional relaxation. Combined with the work of R{ö}dl, Schacht, Siggers, and Tokushige from 2007, this also shows that such hypergraphs admit approximate -decompositions.

Paper Structure

This paper contains 11 sections, 13 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. On Montgomery's Proof and the Difficulties of Generalizing to Hypergraphs
  3. On the Novelty of Our Proof
  4. Organization of the Paper
  5. Missing One Edge and a Fixing Lemma
  6. Concatenation Lemma
  7. Missing One Edge and Fixing
  8. Missing Matchings
  9. A Sampling Lemma and the Proof of Main Result
  10. A Sampling Lemma
  11. Proof of Main Result

Key Result

Theorem 1.2

For each integer $r\ge 3$, there exists a constant $C > 0$ such that the following holds for all integers $q > r$ and $n$ sufficiently large: If $G$ is a $K_q^r$-divisible graph on $n$ vertices with $\delta_{r-1}(G)\ge \left(1-\frac{C}{q^{2r}}\right)n$, then $G$ admits a $K_q^r$-decomposition.

Theorems & Definitions (29)

  • Conjecture 1.1: Glock, Kühn, Osthus GKO20Survey 2020
  • Theorem 1.2: Glock, Kühn, Lo, and Osthus GKLO16 2016
  • Theorem 1.3: Barber, Kühn, Lo, Montgomery, and Osthus BKLMO2017fractional 2017
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: Concatenation
  • proof
  • Theorem 2.4
  • ...and 19 more