Phase transition of degenerate Tur\'{a}n problems in $p$-norms

TL;DR

This paper investigates the phase transition in the maximum $p$-norm of $F$-free graphs, extending results to hypergraphs and confirming conjectures about precise bounds for specific bipartite graphs.

Contribution

It extends the phase transition analysis of $p$-norm extremal problems from graphs to hypergraphs and refines bounds for certain bipartite graphs.

Findings

Identified the threshold $p_F$ for phase transition in hypergraphs.

Provided the correct constant factor for $p > p_F$ in graphs.

Confirmed the conjecture removing the $ ext{log } n$ factor for specific bipartite graphs.

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. F\"{u}redi and K\"{u}ndgen \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 $p = p_F$, F\"{u}redi and K\"{u}ndgen 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.