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Density Hajnal--Szemerédi theorem for cliques of size four

Jianfeng Hou, Caiyun Hu, Xizhi Liu, Yixiao Zhang

TL;DR

This paper advances the density version of the Hajnal–Szemerédi theorem for K4 by asymptotically identifying five extremal construction classes and proposing a set of candidates for general r≥5. It develops a multi-layered framework that partitions a maximum K4-tiling into six A-subfamilies and uses intricate local and global edge-estimation arguments, including rotations and switching paths, combined with convex optimization, to bound the total edge count. The main contribution is a tight asymptotic formula ex$(n,(k+1)K_4)= ext{Xi}(n,k)+O(n)$ across several regimes of k/n, together with a stability version and a detailed program for general r≥5. The methods blend classical extremal tools (Erdős–Gallai, Simonovits, Zarankiewicz) with novel partitioning, switching, and convexity-based optimization to control edge densities and confirm the extremal role of the proposed constructions.

Abstract

The celebrated Corrádi--Hajnal Theorem~\cite{CH63} and the Hajnal--Szemerédi Theorem~\cite{HS70} determined the exact minimum degree thresholds for a graph on $n$ vertices to contain $k$ vertex-disjoint copies of $K_r$, for $r=3$ and general $r \ge 4$, respectively. The edge density version of the Corrádi--Hajnal Theorem was established by Allen--Böttcher--Hladký--Piguet~\cite{ABHP15} for large $n$. Remarkably, they determined the four classes of extremal constructions corresponding to different intervals of $k$. They further proposed the natural problem of establishing a density version of the Hajnal--Szemerédi Theorem: For $r \ge 4$, what is the edge density threshold that guarantees a graph on $n$ vertices contains $k$ vertex-disjoint copies of $K_r$ for $k \le n/r$. They also remarked, ``We are not even sure what the complete family of extremal graphs should be.'' We take the first step toward this problem by determining asymptotically the five classes of extremal constructions for $r=4$. Furthermore, we propose a candidate set comprising $r+1$ classes of extremal constructions for general $r \ge 5$.

Density Hajnal--Szemerédi theorem for cliques of size four

TL;DR

This paper advances the density version of the Hajnal–Szemerédi theorem for K4 by asymptotically identifying five extremal construction classes and proposing a set of candidates for general r≥5. It develops a multi-layered framework that partitions a maximum K4-tiling into six A-subfamilies and uses intricate local and global edge-estimation arguments, including rotations and switching paths, combined with convex optimization, to bound the total edge count. The main contribution is a tight asymptotic formula ex across several regimes of k/n, together with a stability version and a detailed program for general r≥5. The methods blend classical extremal tools (Erdős–Gallai, Simonovits, Zarankiewicz) with novel partitioning, switching, and convexity-based optimization to control edge densities and confirm the extremal role of the proposed constructions.

Abstract

The celebrated Corrádi--Hajnal Theorem~\cite{CH63} and the Hajnal--Szemerédi Theorem~\cite{HS70} determined the exact minimum degree thresholds for a graph on vertices to contain vertex-disjoint copies of , for and general , respectively. The edge density version of the Corrádi--Hajnal Theorem was established by Allen--Böttcher--Hladký--Piguet~\cite{ABHP15} for large . Remarkably, they determined the four classes of extremal constructions corresponding to different intervals of . They further proposed the natural problem of establishing a density version of the Hajnal--Szemerédi Theorem: For , what is the edge density threshold that guarantees a graph on vertices contains vertex-disjoint copies of for . They also remarked, ``We are not even sure what the complete family of extremal graphs should be.'' We take the first step toward this problem by determining asymptotically the five classes of extremal constructions for . Furthermore, we propose a candidate set comprising classes of extremal constructions for general .
Paper Structure (27 sections, 62 theorems, 381 equations, 49 figures, 2 tables)

This paper contains 27 sections, 62 theorems, 381 equations, 49 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Extremal constructions
  4. Setup
  5. Preliminaries
  6. Local estimation i: basic upper bounds
  7. Upper bounds for $e(\mathcal{A}_i, \mathcal{A}_j)$
  8. Upper bounds for $e(\mathcal{A}_i, \mathcal{B}\cup \mathcal{C} \cup \mathcal{D})$
  9. Local estimation ii$\colon$ $e(\mathcal{A}_3)$
  10. Local estimation iii$\colon$ $e(\mathcal{A}_4) + e(\mathcal{A}_4, \mathcal{B} \cup \mathcal{C})$
  11. $\mathcal{A}_{4}$ and $\mathcal{B}$
  12. $\mathcal{A}_{4}$ and $\mathcal{C}$
  13. Local estimation iv$\colon$ $e(\mathcal{A}_6 \cup \mathcal{B} \cup \mathcal{C} \cup \mathcal{D})$
  14. Proof of Lemma \ref{['LEMMA:A6BCD-upper-bound-b']}
  15. Preparations for Lemma \ref{['LEMMA:A6BCD-upper-bound-a']} i: an extension of Turán
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $r \ge 3$ and $n \ge r (k+1) \ge 0$ be integers. Suppose that Then $\nu(K_{r}, G) \ge k + 1$.

Figures (49)

  • Figure 1: The asymptotic behavior of $\frac{\mathrm{ex}(n,(k+1)K_4)}{n^2}$ as a function of $\frac{k}{n}$.
  • Figure 2: Structures of $E_{1}(n,k), \ldots, E_{5}(n,k)$.
  • Figure 3: Members in $\mathcal{A}_{1}, \ldots, \mathcal{A}_{4}$.
  • Figure 4: Structure of the proof for Theorem \ref{['THM:Mian-HS-density-K4']}. Here, $\alpha \coloneqq \frac{20+\sqrt{10}}{130} = 0.178171...$.
  • Figure 5: Left: $Q$ is $3$-seen by a triangle (blue) in $\mathcal{B}$, and $Q_i$ is seen by a vertex (red) in $\mathcal{D}$. Right: after rotation, the number of vertex-disjoint copies of $K_4$ increases by one.
  • ...and 44 more figures

Theorems & Definitions (165)

  • Theorem 1.1: Corrádi--Hajnal CH63, Hajnal--Szemerédi HS70
  • Theorem 1.2: Allen--Böttcher--Hladký--Piguet ABHP15
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: Erdős--Gallai EG59
  • Theorem 2.2: Erdős--Gallai EG59
  • Theorem 2.3: Simonovits Sim74
  • Theorem 2.5: Gyárfás--Rousseau--Schelp GRS84
  • Theorem 2.6: Čulík Cu56
  • Lemma 3.2
  • ...and 155 more