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On the Turán number of the expansion of the $t$-fan

Xin Cheng, Dániel Gerbner, Hilal Hama Karim, Junpeng Zhou

TL;DR

This work determines the Turán number for the 3-expansion of the $t$-fan, proving that $ex_3(n,F_t^3)=inom{n}{3}-inom{n-t}{3}$ for large $n$ and $t>1$. The authors develop a framework around heavy edges, partial expansions, and nice star configurations to obtain a sharp upper bound, complemented by a natural lower-bound construction. A stability statement follows, showing extremal hypergraphs are close to the canonical construction. The paper also sketches extensions to related expansions and higher uniformities, placing the result in the broader context of hypergraph Turán problems and expansions.

Abstract

The $t$-fan is the graph on $2t+1$ vertices consisting of $t$ triangles which intersect at exactly one common vertex. For a given graph $F$, the $r$-expansion $F^r$ of $F$ is the $r$-uniform hypergraph obtained from $F$ by adding $r-2$ distinct new vertices to each edge of $F$. We determine the Turán number of the 3-expansion of the $t$-fan for sufficiently large $n$.

On the Turán number of the expansion of the $t$-fan

TL;DR

This work determines the Turán number for the 3-expansion of the -fan, proving that for large and . The authors develop a framework around heavy edges, partial expansions, and nice star configurations to obtain a sharp upper bound, complemented by a natural lower-bound construction. A stability statement follows, showing extremal hypergraphs are close to the canonical construction. The paper also sketches extensions to related expansions and higher uniformities, placing the result in the broader context of hypergraph Turán problems and expansions.

Abstract

The -fan is the graph on vertices consisting of triangles which intersect at exactly one common vertex. For a given graph , the -expansion of is the -uniform hypergraph obtained from by adding distinct new vertices to each edge of . We determine the Turán number of the 3-expansion of the -fan for sufficiently large .
Paper Structure (4 sections, 6 theorems, 1 equation)

This paper contains 4 sections, 6 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thmnew1']}
  4. Concluding remarks

Key Result

Theorem 1.1

For any $t>1$, if $n$ is sufficiently large, then $\mathrm{ex}_3(n,F_t^3)=\binom{n}{3}-\binom{n-t}{3}$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Lemma 2.1
  • proof : Proof
  • Lemma 2.2
  • proof : Proof
  • Proposition 2.3
  • proof
  • proof : Proof of Theorem \ref{['thmnew1']}
  • Claim 3.1
  • proof : Proof of Claim
  • ...and 14 more