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On generalized Turán problems for expansions

Junpeng Zhou, Xiamiao Zhao, Xiying Yuan

TL;DR

We study generalized Turán numbers ${\rm ex}_r(n, K_s^{(r)}, F^r)$ for the $r$-uniform expansions $F^r$ of graphs, distinguishing non-degenerate cases (when ${\chi(F)>s}$) from degenerate ones. The non-degenerate regime yields exact results for expansions of complete graphs, with ${\rm ex}_r(n, K_s^{(r)}, K_{\\ell+1}^r) = {\\mathcal N}(K_s^{(r)}, \mathcal{T}_r(n,\\ell))$ for large $n$, and asymptotics for the vertex-disjoint union of complete graphs. In the degenerate regime, the authors determine asymptotics for expansions of star forests, linear forests, and star-path forests, including sharp $O(n^{r-2})$ bounds and exact asymptotics in broad ranges. A key methodological thread is relating hypergraph results to graph Turán numbers via bounds like ${\rm ex}_r(n, K_s^{(r)}, F^r) \le {\rm ex}_s(n, F^s)$ and the Berge-${\mathcal F}$ framework, enabling transfer of classical results to hypergraph expansions. Overall, the work advances systematic understanding of generalized Turán problems for expansions and provides a suite of exact and asymptotic results across non-degenerate and degenerate settings.

Abstract

Given a graph $F$, the $r$-expansion $F^r$ of $F$ is the $r$-uniform hypergraph obtained from $F$ by inserting $r-2$ new distinct vertices in each edge of $F$. Given $r$-uniform hypergraphs $\cH$ and $\cF$, the generalized Turán number, denoted by $\ex_r(n,\cH,\cF)$, is the maximum number of copies of $\cH$ in an $n$-vertex $r$-uniform hypergraph that does not contain $\cF$ as a subhypergraph. In the case where $r=2$ (i.e., the graph case), the study of generalized Turán problems was initiated by Alon and Shikhelman [\textit{J. Combin. Theory Series B.} 121 (2016) 146--172]. Motivated by their work, we systematically study generalized Turán problems for expansions and obtain several general and exact results. In particular, for the non-degenerate case, we determine the exact generalized Turán number for expansions of complete graphs, and establish the asymptotics of the generalized Turán number for expansions of the vertex-disjoint union of complete graphs. For the degenerate case, we establish the asymptotics of generalized Turán numbers for expansions of several classes of forests, including star forests, linear forests and star-path forests.

On generalized Turán problems for expansions

TL;DR

We study generalized Turán numbers for the -uniform expansions of graphs, distinguishing non-degenerate cases (when ) from degenerate ones. The non-degenerate regime yields exact results for expansions of complete graphs, with for large , and asymptotics for the vertex-disjoint union of complete graphs. In the degenerate regime, the authors determine asymptotics for expansions of star forests, linear forests, and star-path forests, including sharp bounds and exact asymptotics in broad ranges. A key methodological thread is relating hypergraph results to graph Turán numbers via bounds like and the Berge- framework, enabling transfer of classical results to hypergraph expansions. Overall, the work advances systematic understanding of generalized Turán problems for expansions and provides a suite of exact and asymptotic results across non-degenerate and degenerate settings.

Abstract

Given a graph , the -expansion of is the -uniform hypergraph obtained from by inserting new distinct vertices in each edge of . Given -uniform hypergraphs and , the generalized Turán number, denoted by , is the maximum number of copies of in an -vertex -uniform hypergraph that does not contain as a subhypergraph. In the case where (i.e., the graph case), the study of generalized Turán problems was initiated by Alon and Shikhelman [\textit{J. Combin. Theory Series B.} 121 (2016) 146--172]. Motivated by their work, we systematically study generalized Turán problems for expansions and obtain several general and exact results. In particular, for the non-degenerate case, we determine the exact generalized Turán number for expansions of complete graphs, and establish the asymptotics of the generalized Turán number for expansions of the vertex-disjoint union of complete graphs. For the degenerate case, we establish the asymptotics of generalized Turán numbers for expansions of several classes of forests, including star forests, linear forests and star-path forests.
Paper Structure (11 sections, 33 theorems, 110 equations)

This paper contains 11 sections, 33 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. The non-degenerate case
  4. The degenerate case
  5. Proofs of Proposition \ref{['prop2.1']}, Theorems \ref{['thm0']} and \ref{['thm00']}
  6. Proofs of Theorems \ref{['thm1']} and \ref{['thm2']}
  7. Proofs of Theorems \ref{['thm1.1']} and \ref{['thm3']}
  8. Concluding remarks
  9. Summary of main results
  10. A general bound on $\mathrm{ex}_r(n,K_s^{(r)},{\mathcal{F}})$
  11. Proof of Theorem \ref{['Zhou-Yuan']}

Key Result

Theorem 1.1

Let integers $\ell \geq r\geq 3$. If $n$ is sufficiently large, then $\mathrm{ex}_r(n, K_{\ell+1}^{r}) =t_r(n,\ell)$ and ${\mathcal{T}}_r(n,\ell)$ is the unique extremal hypergraph.

Theorems & Definitions (49)

  • Theorem 1.1: Pikhurko Pi
  • Theorem 1.2: Gerbner Ge2
  • Theorem 1.3: Duke and Erdős AA4
  • Conjecture 1.4: Erdős Matching Conjecture Er1
  • Theorem 1.5: Bushaw and Kettle BK
  • Theorem 1.6: Khormali and Palmer KP
  • Theorem 1.7: Chakraborti and Chen ChCh
  • Theorem 1.8: Liu and Wang LiWa
  • Theorem 1.9: Gerbner Gerb1
  • Theorem 1.10: Axenovich, Gerbner, Liu and Patkós AGLP
  • ...and 39 more