Leírás
A subfamily $\{F_1,F_2,\dots,F_{|P|}\}\subseteq \mathcal{F}$ is a copy of the poset $P$ if there exists a bijection $i:P\rightarrow \{F_1,F_2,\dots,F_{|P|}\}$ such that $p\le_P q$ implies $i(p)\subseteq i(q)$. A family $\mathcal{F}$ is $P$-free, if it does not contain any copy of $P$. For any values of $k$ and $l$ we construct a $\{\wedge_k,\vee_l\}$-free family that we conjecture to contain asymptotically the maximum number of pairs in containment. We prove the conjecture for some small values of $k$ and $l$ and we show that the conjecture holds under the stronger condition that the families considered are also $P_4$-free. We also derive the asymptotics of the maximum number of copies of certain tree posets $T$ of height 2 in $\{\wedge_k,\vee_l\}$-free families $\mathcal{F}\subseteq 2^{[n]}$. Joint work with D. Gerbner, A. Methuku, D. Nagy, and M. Mizer.