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 )for any fixed m ≥ 1. In this paper we show that La(n, P ) ≤ 1 2 k−1 |P | + (3k − 5)2 k−2 (h(P ) − 1) − 1 n n/2 for any fixed k ≥ 2, improving the best known upper bound. By choosing k appropriately, we obtain that La(n, P ) = O h(P ) log 2 |P | h(P ) + 2 n n/2 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 O(n c ) for every c > 1 2 .

Let F = (U, E) be a graph and H = (V, E) be a hypergraph. We say that H contains a Berge-F if there exist injections ψ : U → V and ϕ : E → E such that for every e = {u, v} ∈ E, {ψ(u), ψ(v)} ⊂ ϕ(e). Let ex r (n, F ) denote the maximum number of hyperedges in an r-uniform hypergraph on n vertices which does not contain a Berge-F .For small enough r and non-bipartite F , ex r (n, F ) = Ω(n 2 ); we show that for sufficiently large r, ex r (n, F ) = o(n 2 ). Let th(F ) = min{r 0 : ex r (n, F ) = o(n 2 ) for all r ≥ r 0 }. We show lower and upper bounds for th(F ), the uniformity threshold of F . In particular, we obtain that th(△) = 5, improving a result of Győri [5].We also study the analogous problem for linear hypergraphs. Let ex L r (n, F ) denote the maximum number of hyperedges in an r-uniform linear hypergraph on n vertices which does not contain a Berge-F , and let the linear unformity threshold th L (F ) = min{r 0 : ex L r (n, F ) = o(n 2 ) for all r ≥ r 0 }. We show that th L (F ) is equal to the chromatic number of F .

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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. The diamond poset, denoted Q2, is defined on four elements x, y, z, w with the relations x < y, z and y, z < w. La(n, P ) has been studied for many posets; one of the major open problems is determining La(n, Q2). It is conjectured that La(n, Q2) = (2 + o(1)) n ⌊n/2⌋ , and infinitely many significantly different, asymptotically tight constructions are known.Studying the average number of sets from a family of subsets of [n] on a maximal chain in the Boolean lattice 2 [n] has been a fruitful method. We use a partitioning of the maximal chains and introduce an induction method to show that La(n, Q2) ≤ (2.20711 + o(1)) n ⌊n/2⌋ , improving on the earlier bound of (2.25 + o(1)) n ⌊n/2⌋

The poset Ramsey number R(Q m , Q n ) is the smallest integer N such that any bluered coloring of the elements of the Boolean lattice Q N has a blue induced copy of Q m or a red induced copy of Q n . The weak poset Ramsey number R w (Q m , Q n ) is defined analogously, with weak copies instead of induced copies. It is easy to see that and Thompson [7] improved the upper bound to 5 3 n + 2. In this paper, we solve this problem asymptotically by showing that Axenovich and WalzerIn the diagonal case, Cox and Stolee [6] proved R w (Q n , Q n ) ≥ 2n+1 using a probabilistic construction. In the induced case, Bohman and Peng [2] showed R(Q n , Q n ) ≥ 2n+1 using an explicit construction. Improving these results, we show that R w (Q m , Q n ) ≥ n + m + 1 for all m ≥ 2 and large n by giving an explicit construction; in particular, we prove that R w (Q 2 , Q n ) = n + 3.