Gerbner, DanielMethuku, AbhishekNagy, Daniel T.Patkos, BalazsVizer, Mate2020-01-292020-01-292020-01-292019-11-0110.1007/s00373-019-02094-3https://infoscience.epfl.ch/handle/20.500.14299/164980WOS:000507467000018A subfamily {F-1, F-2, ..., F-vertical bar P vertical bar} subset of F is a copy of the poset P if there exists a bijection i : P -> {F-1, F-2, ..., F-vertical bar P vertical bar}, such that p <= (P) q implies i (p) subset of i (q). A family F is P-free, if it does not contain a copy of P. In this paper we establish basic results on the maximum number of k-chains in a P-free family F subset of 2([n]). We prove that if the height of P, h(P) > k, then this number is of the order Theta(Pi(k+1)(i=1) ((li-1)(li))), where l(0) = n and l(1) >= l(2) >= ... >= l(k+1) are such that n - l(1), l(1) - l(2), ..., l(k) - l(k+1), l(k+1) differ by at most one. On the other hand if h(P) <= k, then we show that this number is of smaller order of magnitude. Let boolean OR(r) denote the poset on r+1 elements a, b(1), b(2), ..., b(r), where a < b(i) for all 1 <= i <= r and let. r denote its dual. For any values of k and l, we construct a {boolean AND(k), boolean OR(l)}-free family and we conjecture that it contains asymptotically the maximum number of pairs in containment. We prove that this conjecture holds under the additional assumption that a chain of length 4 is forbidden. Moreover, we prove the conjecture for some small values of k and l. We also derive the asymptotics of the maximum number of copies of certain tree posets T of height 2 in {boolean AND(k), boolean OR(l)}-free families F subset of 2([n]).Mathematicsforbidden subposet problemscomparable pairsset systemstext::journal::journal article::research article