LAUSR

Let La(n, P) be the maximum size of a family of subsets of [n] = {1, 2, … , n} not containing P as a (weak) subposet, and let h(P) be the length of a longest chain in P. The best known upper bound for La(n, P) in terms of |P| and h(P) is due to Chen and Li, who showed that La(n,P)≤1m+1|P|+12(m2+3m−2)(h(P)−1)−1n⌊n/2⌋$\text {La}(n,P) \le \frac {1}{m+1} \left (|{P}| + \frac {1}{2}(m^{2} +3m-2)(h(P)-1) -1 \right ) {\left (\begin {array}{c}{n}\\ {\lfloor n/2 \rfloor } \end {array}\right )}$ for any fixed m ≥ 1. In this paper we show that La(n,P)≤12k−1|P|+(3k−5)2k−2(h(P)−1)−1n⌊n/2⌋$\text {La}(n,P) \le \frac {1}{2^{k-1}} \left (|P| + (3k-5)2^{k-2}(h(P)-1) - 1 \right ) {\left (\begin {array}{c}{n}\\ {\lfloor n/2 \rfloor } \end {array}\right )}$ for any fixed k ≥ 2, improving the best known upper bound. By choosing k appropriately, we obtain that La(n,P)=????h(P)log2|P|h(P)+2n⌊n/2⌋$\text {La}(n,P) = \mathcal {O}\left (h(P) \log _{2}\left (\frac {|{P}|}{h(P)}+2\right ) \right ) {\left (\begin {array}{c}{n}\\ {\lfloor n/2 \rfloor } \end {array}\right )}$ as a corollary, which we show is best possible for general P. We also give a different proof of this corollary by using bounds for generalized diamonds. We also show that the Lubell function of a family of subsets of [n] not containing P as an induced subposet is ????(nc)$\mathcal {O}(n^{c})$ for every c>12$c>\frac {1}{2}$.