Phase Transition of Degenerate Turán Problems in \({p}\)-Norms

.For a positive real number \(p\), the \(p\)-norm \( {G} _p\) of a graph \(G\) is the sum of the \(p\)th powers of all vertex degrees. We study the maximum \(p\)-norm \(exₚ(n,F)\) of \(F\)-free graphs on \(n\) vertices. Füredi and Kündgen [J. Graph Theory, 51 (2006), pp. 37–48] showed that for every bipartite graph \(F\), there exists a threshold \(p_F\) such that for \(p< pF\), the order of \(exₚ(n,F)\) is governed by pseudorandom constructions, while for \(p > pF\), it is governed by star-like constructions, assuming a mild assumption on the growth rate of \(ex(n,F)\). The main contribution of our paper is extending this result to hypergraphs. Moreover, in the case of graphs, our proof differs from that in [Z. Füredi and A. Kündgen, J. Graph Theory, 51 (2006), pp. 37–48], offering the advantage of producing the correct constant factor when \(p > pF\). When \(p=p_F\), Füredi and Kündgen proved a general upper bound on \(exₚ(n,F)\) that is 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-sided degree-bounded graphs that meet Füredi's bound and families of short even cycles.Keywordsdegenerate Turán problemdegree powerscounting starsphase transitionMSC codes05C0705C35