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We show that every complete n-vertex simple topological graph contains a topological subgraph on at least (logn)1/4−o(1) vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph. This is the first improvement on the bound Ω(log1/8n) obtained in 2003 by Pach, Solymosi, and Tóth. We also show that every complete n-vertex simple topological graph contains a plane path of length at least (logn)1−o(1) . 

We show that there is an absolute constant c > 0 c>0 such that the following holds. For every n > 1 n > 1 , there is a 5-uniform hypergraph on at least 2 2 c n 1 / 4 2^{2^{cn^{1/4}}} vertices with independence number at most n n , where every set of 6 vertices induces at most 3 edges. The double exponential growth rate for the number of vertices is sharp. By applying a stepping-up lemma established by the first two authors, analogous sharp results are proved for k k -uniform hypergraphs. This answers the penultimate open case of a conjecture in Ramsey theory posed by Erdős and Hajnal in 1972. 

Given a finite point set P in the plane, a subset S⊆P is called an island in P if conv(S) ⋂ P = S . We say that S ⊂ P is a visible island if the points in S are pairwise visible and S is an island in P. The famous Big-line Big-clique Conjecture states that for any k ≥ 3 and l ≥ 4, there is an integer n = n(k, l ), such that every finite set of at least n points in the plane contains l collinear points or k pairwise visible points. In this paper, we show that this conjecture is false for visible islands, by replacing each point in a Horton set by a triple of collinear points. Hence, there are arbitrarily large finite point sets in the plane with no 4 collinear members and no visible island of size 13. 

Given a family F of k-element sets, S₁,…,S_r ∈ F form an r-sunflower if S_i ∩ S_j = S_{i'} ∩ S_{j'} for all i ≠ j and i' ≠ j'. According to a famous conjecture of Erdős and Rado (1960), there is a constant c = c(r) such that if |F| ≥ c^k, then F contains an r-sunflower. We come close to proving this conjecture for families of bounded Vapnik-Chervonenkis dimension, VC-dim(F) ≤ d. In this case, we show that r-sunflowers exist under the slightly stronger assumption |F| ≥ 2^{10k(dr)^{2log^{*} k}}. Here, log^* denotes the iterated logarithm function. We also verify the Erdős-Rado conjecture for families F of bounded Littlestone dimension and for some geometrically defined set systems.