给定正整数q≥p≥r≥2以及一个r-一致超图H,如果对于H的任意一个q-顶点子集Q ⊆ V(H),均存在一个p-顶点子集P ⊆ Q使得P在超图H中的导出子图是一个团,则称超图 H 具有性质(q,p).令 Tr(n,q,p)=min{e(H):E(H)⊆([nr]),H具有性质(q,p)}.称 tr(q,p)=limn→∞ Tr(n,q,p)/(nr)为r-一致超图中关于性质(q,p)的局部Turán密度.Frankl等(2021)证明了对于任意正整数a有limp→∞tr(ap+1,p+1)=1/ar-1以及对于所有正整数p≥3有t3(2p+1,p+1)=1/4.同时他们提出了对给定实数γ＞1确定极限limp→∞tr(γp+1,p+1)值的问题.基于超图Turán密度的研究,本文给出一些局部Turán密度的准确值,部分地回答了他们的问题.特别地,本文的结论表明他们的问题中关于极限具体值的论断是不成立的.
Abstract
For integers q ≥ p≥r≥2,we say that an r-uniform hypergraph H has property(q,p),if for any q-vertex subset Q of V(H),there exists a p-vertex subset P of Q spanning a clique in H.Let Tr(n,q,p)=min{e(H):E(H)⊆([n]r),H has property(q,p)}.The local Turán density about property(q,p)in r-uniform hypergraphs is defined as tr(q,p)=limn→∞ Tr(n,q,p)/(nr).Frankl et al.(2021)proved that limp→∞ tr(ap+1,p+1)=1/ar-1for any positive integer a and t3(2p+1,p+1)=1/4 for all p ≥ 3 and asked the question on how to determine the value of limp→∞ tr(γp+1,p+1)where γ ≥ 1 is a real number.Based on the study of hypergraph Turán densities,we determine some exact values of local Turán densities and answer their question partially;in particular,our results imply that the statement in their question about the exact value of the limit is false.
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