We show that for every sufficiently large n, the number of monotone subsequences of length four in a permutation on n points is at least. Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colorings of complete graphs with two colors, where the number of monochromatic K 4 's is minimized. We show that all the extremal colorings must contain monochromatic K 4 's only in one of the two colors. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

Graphons are analytic objects associated with convergent sequences of graphs. Problems from extremal combinatorics and theoretical computer science led to a study of graphons determined by finitely many subgraph densities, which are referred to as finitely forcible. Following the intuition that such graphons should have finitary structure, Lovász and Szegedy conjectured that the topological space of typical vertices of a finitely forcible graphon is always compact. We disprove the conjecture by constructing a finitely forcible graphon such that the associated space is not compact. The construction method gives a general framework for constructing finitely forcible graphons with non-trivial properties.

Abstract. We show that for every ε > 0 there exist δ > 0 and n0 ∈ N such that every 3-uniform hypergraph on n ≥ n0 vertices with the property that every k-vertex subset, where k ≥ δn, induces at leastas a subgraph, where K − 4 is the 3-uniform hypergraph on 4 vertices with 3 edges. This question was originally raised by Erdős and Sós. The constant 1/4 is the best possible.

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Erdős and Sós proposed a problem of determining the maximum number F (n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that n and a, b, c, d are as equal as possible. We prove that the conjectured recurrence holds for sufficiently large n. We also prove the conjecture for n = 4 k for all k ≥ 0. These results imply that lim F (n) ( n 3 ) = 0.4, and determine the unique limit object. In the proof we use flag algebras combined with stability arguments.