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On Turán problems for suspension hypergraphs

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

TL;DR

The paper studies Turán numbers for suspensions ${\rm S}^rF$ of graphs, linking higher-uniformity extremal problems to base-graph questions through link-graphs and design-based constructions. It leverages the hypergraph removal lemma to relate ${\rm ex}_r(n,{ m S}^rF)$ to ${\rm ex}_r(n,{ m S}^rK_k)$ for graphs with chromatic number $k$, and proves sharp bounds for specific cases such as ${\rm ex}_4(n,{ m S}^4C_{2k+1})$ and an exact bound for ${\rm ex}_3(n,{ m S}^3(P_3\cup K_2))$ via explicit designs. Key contributions include the general bound ${\rm ex}_r(n,{ m S}^rF) = O(n^{r-2}{\rm ex}(n,F))$, asymptotic reductions to $K_k$-suspensions, and concrete sharp results in the 3- and 4-uniform settings that highlight the role of suspensions in Turán-type questions. These results advance the understanding of suspension-induced extremal behavior and connect to design theory and sunflower-type phenomena in hypergraphs.

Abstract

For a given graph $F$, the $r$-uniform suspension of $F$ is the $r$-uniform hypergraph obtained from $F$ by taking $r-2$ new vertices and adding them to every edge. In this paper, we consider Turán problems on suspension hypergraphs, and we obtain several general and exact results.

On Turán problems for suspension hypergraphs

TL;DR

The paper studies Turán numbers for suspensions of graphs, linking higher-uniformity extremal problems to base-graph questions through link-graphs and design-based constructions. It leverages the hypergraph removal lemma to relate to for graphs with chromatic number , and proves sharp bounds for specific cases such as and an exact bound for via explicit designs. Key contributions include the general bound , asymptotic reductions to -suspensions, and concrete sharp results in the 3- and 4-uniform settings that highlight the role of suspensions in Turán-type questions. These results advance the understanding of suspension-induced extremal behavior and connect to design theory and sunflower-type phenomena in hypergraphs.

Abstract

For a given graph , the -uniform suspension of is the -uniform hypergraph obtained from by taking new vertices and adding them to every edge. In this paper, we consider Turán problems on suspension hypergraphs, and we obtain several general and exact results.

Paper Structure

This paper contains 3 sections, 8 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Proofs of Theorems \ref{['asy']} and \ref{['shar']}
  3. Proofs of Proposition \ref{['simpli']} and Theorem \ref{['fores']}

Key Result

Proposition 1.1

Let $k<r$. For any graph $F$ and $n$, we have $\mathrm{ex}_r(n,{\mathcal{S}}^rF)\le \binom{n}{r-k}\mathrm{ex}_k(n-r+k,{\mathcal{S}}^kF)/\binom{r}{k}$. In particular, $\mathrm{ex}_r(n,{\mathcal{S}}^rF)\le \binom{n}{r-2}\mathrm{ex}(n-r+2,F)/\binom{r}{2}=(1+o(1))2\binom{n}{r}\mathrm{ex}(n,F)/n^2$.

Theorems & Definitions (25)

  • Proposition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof : Proof
  • proof : Proof of Theorem \ref{['asy']}
  • Proposition 2.2
  • proof : Proof
  • ...and 15 more