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Tiling $H$ in dense graphs

Nannan Chen, Xizhi Liu, Lin Sun, Guanghui Wang

TL;DR

This work determines the asymptotic extremal behavior for tiling the $H$-shaped tree in dense $r$-graphs, identifying two near-extremal constructions and showing the first resembles the complement of two cliques, thereby refuting Lang's conjecture in general. The authors develop a framework combining the Regularity Lemma and Blow-up Lemma, reducing the problem to tiling a dense reduced graph with $K_2$, $H$, and $ ilde{H}$, and then lifting this tiling to the original graph to obtain a large $H$-tiling. They prove a sharp asymptotic for $ ext{ex}(n,eta n ext{·}H)$: $ ext{ex}(n,eta n ext{·}H) = (oldsymbol{ar{ ext{Ξ}}}(eta) + o(1)) n^2$ with $oldsymbol{ar{ ext{Ξ}}}(eta) = 3eta(1-3eta)$ for $0\leeta\le 1/9$ and $18eta^2$ for $1/9\leeta\\le 1/6$, and the extremal example for the low-$eta$ range is the complete bipartite graph with parts of sizes approximately $3eta n$ and $(1-3eta)n$. Beyond the main result, the paper introduces a refined dichotomy between rigid and nonrigid spanning subgraphs, proposing a revised Lang-type conjecture and highlighting open questions about density tiling, particularly for trees, with implications for longstanding conjectures in extremal combinatorics.

Abstract

We determine asymptotically the two extremal constructions for the tiling problem of the $H$-shaped tree. In particular, the first extremal construction is close to the complement of two cliques, in contrast to previously studied bipartite graphs, where the first extremal construction is close to the complement of a single clique. This result refutes one of Lang's conjectures [arXiv:2308.12281], which seeks to generalize the Erdős Matching Conjecture.

Tiling $H$ in dense graphs

TL;DR

This work determines the asymptotic extremal behavior for tiling the -shaped tree in dense -graphs, identifying two near-extremal constructions and showing the first resembles the complement of two cliques, thereby refuting Lang's conjecture in general. The authors develop a framework combining the Regularity Lemma and Blow-up Lemma, reducing the problem to tiling a dense reduced graph with , , and , and then lifting this tiling to the original graph to obtain a large -tiling. They prove a sharp asymptotic for : with for and for , and the extremal example for the low- range is the complete bipartite graph with parts of sizes approximately and . Beyond the main result, the paper introduces a refined dichotomy between rigid and nonrigid spanning subgraphs, proposing a revised Lang-type conjecture and highlighting open questions about density tiling, particularly for trees, with implications for longstanding conjectures in extremal combinatorics.

Abstract

We determine asymptotically the two extremal constructions for the tiling problem of the -shaped tree. In particular, the first extremal construction is close to the complement of two cliques, in contrast to previously studied bipartite graphs, where the first extremal construction is close to the complement of a single clique. This result refutes one of Lang's conjectures [arXiv:2308.12281], which seeks to generalize the Erdős Matching Conjecture.
Paper Structure (11 sections, 20 theorems, 67 equations, 64 figures)

This paper contains 11 sections, 20 theorems, 67 equations, 64 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['THM:Main-H-tiling']}
  4. Proof of Proposition \ref{['PROP:H-K2-H-hat-tiling']}
  5. Local estimation
  6. Upper bound for $e(\mathcal{H}_0, \mathcal{H}_1)$
  7. Upper bounds for $e(\mathcal{H}_1)$, $e(\mathcal{H}_1, \mathcal{H}_2)$, and $e(\mathcal{H}_1, \mathcal{H}_3)$
  8. Upper bound for $e(\mathcal{H}_2, \mathcal{H}_i)$ for $i \in \{0,1,2\}$
  9. Upper bound for $e(\mathcal{H}_2, \mathcal{H}_3)$
  10. Upper bound for $e(\mathcal{H}_3)$
  11. Concluding remarks

Key Result

Theorem 1.2

Let $t \ge s \ge 1$ be integers. Suppose that $F$ is a spanning subgraph of $K_{s,t}$. Then, for every real number $\beta \in \left(0,\frac{1}{s+t}\right)$,

Figures (64)

  • Figure 1: $H$ and $\hat{H}$.
  • Figure 2: The asymptotic behavior of $\frac{\mathrm{ex}(n,\beta n \cdot H)}{n^2}$ as a function of $\beta$. The red dotted curve represents the conjectured value.
  • Figure 3: Five ways to embed $H$ into a blowup of $\hat{H}$.
  • Figure 4: Auxiliary figure for the proof of Claim \ref{['CLAIM:one-edge-b']}\ref{['CLAIM:one-edge-b-3']}.
  • Figure 5: Decomposition of $V(H_i \cup H_j) \cup \{w\}$ into $\hat{H}$ and three pairwise disjoint edges.
  • ...and 59 more figures

Theorems & Definitions (128)

  • Conjecture 1.1: Lang Lang23
  • Theorem 1.2: Grosu--Hladký GH12
  • Theorem 1.3
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4: KSSS02
  • Lemma 2.5: GH12
  • Lemma 2.6: GH12
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['PROP:K2-H-hat-decomposition']}
  • ...and 118 more