r-一致超图Ramsey函数的渐近下界(英文)

Asymptotic Lower Bounds of Ramsey Numbers for r-uniform Hypergraphs

本文利用Lovasz局部引理的Spencer形式和对称形式给出r-一致超图Ramsey函数的渐近下界.证明了:对于任意取定的正整数f<k和m≥r+1≥3,存在常数c=c(l,m)>0,使得当n→∞时,有R^((r))(m^l,n^(k-l))≥(c-o(1))(n^(r-1)/logn)~■.特别地,R^((r))_k(n)≥(1-o(1))n/e k~■(n→∞).对于任意取定的正整数s≥r+1和常数δ>0,α≥0,如果F表示阶为s的r-一致超图,■表示阶为t的r-一致超图,且■的边数满足m(■)≥(δ-o(1))t^r/(logt)α(t→∞),则存在c=c(s,δ,α)>0,使得R^((r))(F,■)≥(c-o(1))(t^(r-1)/(logt)~l+(r-l)α)^(m(F)-l/s-r).

In this paper,we apply the Spencer＇s form and symmetric form of Lovaszlocal lemma to derive the asymptotic lower bounds for r-uniform hypergraphs.For any fixedpositive integer lk and m≥r ＋ 1≥3,there exists a constant c = c（r,l,m）0 such thatR^（（r））（m^1,n^（k-1））≥（c-o（1））（n^（r-1）/log n）^（■-1/m-r）（n→∞）.In particular,R_k^（（r））（n）≥（1 - o（1）） n/ek^（■-1/n-r）（n→∞）.And for any fixed integer s≥r ＋ 1,constantsδ0 andα≥0,if F is a runiformhypergraph on s vertices and G is a r-uniform hypergraph on t vertices with m（G）≥（δ-o（1））t^r/（log t）~αas t→∞,then there exists a constant c = c（s,δ,α）0 such that R^（（r））（F,G）≥（c-o（1））（t^（r-1）/（log t^（1＋（r-1）α））^（m（F）-1/s-r）.