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Finite Hypergraph Families with Rich Extremal Turán Constructions via Mixing Patterns

Xizhi Liu, Oleg Pikhurko

TL;DR

This work extends the mixing-pattern framework of prior results to finite families of minimal $r$-graph patterns, proving that maximum ${\mathcal F}_M$-free $r$-graphs are exactly the $P_I$-mixing constructions and that almost-extremal graphs are edit-distance close to such constructions. It delivers two striking applications: (i) a finite 3-graph family with exponentially many maximum ${\mathcal F}$-free graphs for large $n$ and with non-finite stability, and (ii) a finite family whose feasible-region set of shadow- and edge-densities forms a Cantor-type set with positive Hausdorff dimension. The core method combines a strong removal lemma, Lagrangian analysis of patterns, and stability/rigidity arguments to control near-extremal structure and to realize a rich landscape of extremal configurations. Together, these results illuminate the richness of hypergraph Turán configurations and advance understanding of the feasible-region problem in hypergraphs, including explicit Cantor-type phenomena.

Abstract

We prove that, for any finite set of minimal $r$-graph patterns, there is a finite family $\mathcal F$ of forbidden $r$-graphs such that the extremal Turán constructions for $\mathcal F$ are precisely the maximum $r$-graphs obtainable from mixing the given patterns in any way via blowups and recursion. This extends the result by the second author \cite{PI14}, where the above statement was established for a single pattern. We present two applications of this result. First, we construct a finite family $\mathcal F$ of $3$-graphs such that there are exponentially many maximum $\mathcal F$-free $3$-graphs of each large order $n$ and, moreover, the corresponding Turán problem is not finitely stable. Second, we show that there exists a finite family $\mathcal{F}$ of $3$-graphs whose feasible region function attains its maximum on a Cantor-type set of positive Hausdorff dimension.

Finite Hypergraph Families with Rich Extremal Turán Constructions via Mixing Patterns

TL;DR

This work extends the mixing-pattern framework of prior results to finite families of minimal -graph patterns, proving that maximum -free -graphs are exactly the -mixing constructions and that almost-extremal graphs are edit-distance close to such constructions. It delivers two striking applications: (i) a finite 3-graph family with exponentially many maximum -free graphs for large and with non-finite stability, and (ii) a finite family whose feasible-region set of shadow- and edge-densities forms a Cantor-type set with positive Hausdorff dimension. The core method combines a strong removal lemma, Lagrangian analysis of patterns, and stability/rigidity arguments to control near-extremal structure and to realize a rich landscape of extremal configurations. Together, these results illuminate the richness of hypergraph Turán configurations and advance understanding of the feasible-region problem in hypergraphs, including explicit Cantor-type phenomena.

Abstract

We prove that, for any finite set of minimal -graph patterns, there is a finite family of forbidden -graphs such that the extremal Turán constructions for are precisely the maximum -graphs obtainable from mixing the given patterns in any way via blowups and recursion. This extends the result by the second author \cite{PI14}, where the above statement was established for a single pattern. We present two applications of this result. First, we construct a finite family of -graphs such that there are exponentially many maximum -free -graphs of each large order and, moreover, the corresponding Turán problem is not finitely stable. Second, we show that there exists a finite family of -graphs whose feasible region function attains its maximum on a Cantor-type set of positive Hausdorff dimension.
Paper Structure (20 sections, 35 theorems, 159 equations, 5 figures)

This paper contains 20 sections, 35 theorems, 159 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Turán problem
  3. Patterns
  4. Finite families with exponentially many extremal Turán $r$-graphs
  5. Feasible region
  6. Notation
  7. Hypergraphs
  8. Further definitions and results for a single pattern
  9. Mixing patterns
  10. Examples
  11. Proof of Theorem \ref{['THM:mixed-pattern']}
  12. Some additional assumptions and definitions related to $P_I$
  13. Basic properties of $P_I$-mixing constructions
  14. Properties of proper patterns
  15. Key lemmas
  16. ...and 5 more sections

Key Result

Theorem 2

Let $r\ge 3$ and let $P_I=\{P_i: i\in I\}$ be an arbitrary collection of minimal $r$-graph patterns, where the index set $I$ is finite. Then there exists $M\in \mathbb{N}$ such that the following statements hold.

Figures (5)

  • Figure 1: The induced subgraph of $L_{B_{5,3}}(i)$ on vertex set $\{4,5,6,7\}$ is a copy of $K_{2,2}$ for $i\in \{1,2,3\}$.
  • Figure 2: The partition structure of a $\{P_1, P_2\}$-mixing construction $G$ with exactly three levels: the base for level-1 is $P_1$, while the bases for the (unique) recursive parts at levels 2 and 3 are respectively $P_2$ and $P_1$.
  • Figure 3: The tree ${\bf T}_G$ of $G$.
  • Figure 4: The partition structure of a $\{P_1, P_2\}$-mixing construction $G$ with exactly three levels: the base for level-1 is $P_1$, while the bases for the unique recursive parts at level 2 and 3 are $P_1$ and $P_2$ respectively.
  • Figure 5: The tree ${\bf T}_G$ of $G$.

Theorems & Definitions (50)

  • Definition 1: $P_I$-Mixed Constructions
  • Theorem 2
  • Theorem 3
  • Theorem 5
  • Corollary 6
  • Lemma 8: PI14
  • Lemma 9: PI14
  • Lemma 10: PI14
  • Lemma 11
  • Lemma 12
  • ...and 40 more