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Phase transition of degenerate Turán problems in $p$-norms

Jun Gao, Xizhi Liu, Jie Ma, Oleg Pikhurko

Abstract

For a positive real number $p$, the $p$-norm $\left\lVert G \right\rVert_p$ of a graph $G$ is the sum of the $p$-th powers of all vertex degrees. We study the maximum $p$-norm $\mathrm{ex}_{p}(n,F)$ of $F$-free graphs on $n$ vertices. Füredi and Kündgen \cite{FK06} show that for every bipartite graph $F$, there exists a threshold $p_F$ such that for $p< p_{F}$, the order of $\mathrm{ex}_{p}(n,F)$ is governed by pseudorandom constructions, while for $p > p_{F}$, it is governed by star-like constructions, assuming a mild assumption on the growth rate of $\mathrm{ex}(n,F)$. The main contribution of our paper is extending this result to hypergraph. Moreover, in the case of graph, our proof differs from that in \cite{FK06}, offering the advantage of producing the correct constant factor when $p > p_{F}$. When $p = p_F$, Füredi and Kündgen proved a general upper bound on $\mathrm{ex}_{p}(n,F)$, tight up to a $\log n$ factor, and conjectured that this factor is unnecessary. We confirm this conjecture for several well-studied bipartite graphs, including one-side degree-bounded graphs and families of short even cycles.

Phase transition of degenerate Turán problems in $p$-norms

Abstract

For a positive real number , the -norm of a graph is the sum of the -th powers of all vertex degrees. We study the maximum -norm of -free graphs on vertices. Füredi and Kündgen \cite{FK06} show that for every bipartite graph , there exists a threshold such that for , the order of is governed by pseudorandom constructions, while for , it is governed by star-like constructions, assuming a mild assumption on the growth rate of . The main contribution of our paper is extending this result to hypergraph. Moreover, in the case of graph, our proof differs from that in \cite{FK06}, offering the advantage of producing the correct constant factor when . When , Füredi and Kündgen proved a general upper bound on , tight up to a factor, and conjectured that this factor is unnecessary. We confirm this conjecture for several well-studied bipartite graphs, including one-side degree-bounded graphs and families of short even cycles.

Paper Structure

This paper contains 8 sections, 22 theorems, 143 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Regularization under the $p$-norm
  4. Proof of Theorem \ref{['THM:main-r-graph']}
  5. Proof of Theorem \ref{['THM:r-graph-critical-point']}
  6. Proof of Theorem \ref{['THM:graph-critical-point']}
  7. Concluding remarks
  8. Sketch of the Proof of Theorem \ref{['THM:Zaran-threshold']}

Key Result

Theorem 1.2

Let $r \ge 2$ be an integer and $p > 1$ be a real number. Suppose that $\mathcal{F}$ is a degenerate family of $r$-graphs satisfying $\mathrm{ex}(n,\mathcal{F}) = O(n^{1+\alpha})$ for some constant $\alpha \in [r-2, r-1)$. Then there exists a constant $C_{\mathcal{F}}>0$ such that In particular, for $r=2$, we have, for every $p > \frac{1}{1-\alpha}$,

Figures (1)

  • Figure 1: Exponents of $\mathrm{ex}_{p}(n,K_{3,3})$, $\mathrm{ex}_{p}(n,C_{4})$, and $\mathrm{ex}_{p}(n,C_{6})$.

Theorems & Definitions (52)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 2.5
  • Proposition 2.6
  • proof : Proof of Proposition \ref{['PROP:random-sample']}
  • Theorem 2.7: Erdős Erdos64
  • Proposition 2.8
  • proof : Proof of Proposition \ref{['PROP:r-partite-subgp-p-norm']}
  • Proposition 2.9: HHLLYZ23
  • ...and 42 more