Families of Subsets Without a Given Poset in Double Chains and Boolean Lattices

Abstract

Given a finite poset P, the intensively studied quantity La(n, P) denotes the largest size of a family of subsets of [n] not containing P as a weak subposet. Burcsi and Nagy (J. Graph Theory Appl. 1, 40–49 2013) proposed a double-chain method to get an upper bound \({\mathrm La}(n,P)\le \frac {1}{2}(|P|+h-2)\left (\begin {array}{c}n \\ \lfloor {n/2}\rfloor \end {array}\right )\) for any finite poset P of height h. This paper elaborates their double-chain method to obtain a new upper bound $${\mathrm La}(n,P)\le \left( \frac{|P|+h-\alpha(G_{P})-2}{2}\right)\left( \begin{array}{c}n \\ \lfloor{\frac{n}{2}}\rfloor\end{array}\right) $$ for any graded poset P, where α(G P ) denotes the independence number of an auxiliary graph defined in terms of P. This result enables us to find more posets which verify an important conjecture by Griggs and Lu (J. Comb. Theory (Ser. A) 119, 310–322, 2012).