We say that a poset \((Q,\le _{Q})\) contains an induced copy of a poset \((P,\le _P)\) if there is an injective function \(\phi :P\rightarrow Q\) such that for every two \(X,Y\in P,\;\; X\le _P Y\) if and only if \(\phi (X)\le _Q \phi (Y)\) . We denote the Boolean lattice \((2^{[n]},\subseteq )\) by \(Q_n\) . Given a fixed 2-coloring c of a poset P, the poset Erdős-Hajnal number of this colored poset is the smallest integer N such that every 2-coloring of the Boolean lattice \(Q_N\) contains an induced copy of P colored as in c, or a monochromatic induced copy of \(Q_n\) . We present bounds on the poset Erdős-Hajnal number of general colored posets, antichains, chains, and small Boolean lattices. Let the poset Ramsey number \(R(Q_n,Q_n)\) be the least N such that every 2-coloring of \(Q_N\) contains a monochromatic induced copy of \(Q_n\) . As a corollary, we show that \(R(Q_n,Q_n)> 2.02n\) , improving on the best known lower bound \(2n+1\) by Cox and Stolee (Order 35(3), 557–579 2018).
