Ramsey-Tur\'{a}n theory for partially-ordered sets

Abstract

We introduce weak and strong poset Ramsey-Tur\'an numbers for $t$-chains in host poset families, focusing on the Boolean lattice family $\mathcal{B}=\{B_n:n\ge 1\}$. For any poset $P$, we show $\operatorname{RT}(\mathcal{B};n,P,l,t)\le \operatorname{RT}^{\sharp}(\mathcal{B};n,P,l,t)$, with equality when $P$ is a chain. In particular, for $t=1$, $\operatorname{RT}(\mathcal{B};n,C_k,l)=\operatorname{RT}^{\sharp}(\mathcal{B};n,C_k,l)=(k-1)(l-1)$. We also give universal upper bounds for both versions. For fixed $k,l,t$ with $\min\{l-1,k-1\}\ge 1$, we prove $\operatorname{RT}^{\sharp}(\mathcal{B};n,A_k,l,t)=\Theta(n^t)$. More generally, for every non-chain poset $P$, the strong number is $\Theta(n^t)$ for fixed $l,t$. Finally, if $h(P)=r>t$ and $l(n)=\lfloor M_n^\beta\rfloor$ with $0<\beta\le \alpha<1$, then both weak and strong versions admit lower bounds of order $\Omega\!\left(2^{\beta n}n^{-\beta/2}\right)$.