Poset Ramsey Number $$R(P,Q_n)$$. II. N-Shaped Poset

Abstract

Abstract Given partially ordered sets (posets) $$(P, \le _P)$$ ( P , ≤ P ) and $$(P', \le _{P'})$$ ( P ′ , ≤ P ′ ) , we say that $$P'$$ P ′ contains a copy of P if for some injective function $$f:P\rightarrow P'$$ f : P → P ′ and for any $$A, B\in P$$ A , B ∈ P , $$A\le _P B$$ A ≤ P B if and only if $$f(A)\le _{P'} f(B)$$ f ( A ) ≤ P ′ f ( B ) . For any posets P and Q , the poset Ramsey number R ( P , Q ) is the least positive integer N such that no matter how the elements of an N -dimensional Boolean lattice are colored in blue and red, there is either a copy of P with all blue elements or a copy of Q with all red elements. We focus on the poset Ramsey number $$R(P, Q_n)$$ R ( P , Q n ) for a fixed poset P and an n -dimensional Boolean lattice $$Q_n$$ Q n , as n grows large. It is known that $$n+c_1(P) \le R(P,Q_n) \le c_2(P) n$$ n + c 1 ( P ) ≤ R ( P , Q n ) ≤ c 2 ( P ) n , for positive constants $$c_1$$ c 1 and $$c_2$$ c 2 . However, there is no poset P known, for which $$R(P, Q_n)> (1+\epsilon )n$$ R ( P , Q n ) > ( 1 + ϵ ) n , for $$\epsilon >0$$ ϵ > 0 . This paper is devoted to a new method for finding u