超图的局部Turan密度问题

On local Turan density problems of hypergraphs

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T_(r)(n,q,p)=min{e(H):E(H)■(r^(n)),H具有性质(q,p)}.称t_(r)(q,p)=lim_(n→∞)T_(r)(n,q,p)(r^(n))为r-一致超图中关于性质(q,p)的局部Tur′an密度.Frankl等(2021)证明了对于任意正整数a有lim_(p→∞)t_(r)(rp+1,p+1)=1/(a^(r-1))以及对于所有正整数p≥3有t_(3)(2p+1,p+1)=1/4.同时他们提出了对给定实数γ>1确定极限lim_(p→∞)t_(r)(rp+1,p+1)值的问题.基于超图Tur′an密度的研究,本文给出一些局部Tur′an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.

For integers q≥p≥r≥2,we say that an r-uniform hypergraph H has property(q,p),if for any qvertex subset Q of V(H),there exists a p-vertex subset P of Q spanning a clique in H.Let T_(r)(n,q,p)=min{e(H):E(H)■(r^(n)),H has property(q,p).The local Turan density about property(q,p)in r-uniform hypergraphs is defined as t_(r)(q,p)=lim_(n→∞)T_(r)(n,q,p)(r^(n)).Frankl et al.(2021)proved that lim_(p→∞)t_(r)(rp+1,p+1)=1/(a^(r-1))for any positive integer a and t_(3)(2p+1,p+1)=1/4 for all p≥3 and asked the question on how to determine the value of lim_(p→∞)t_(r)(rp+1,p+1) where≥1 is a real number.Based on the study of hypergraph Turan densities,we determine some exact values of local Turan 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.