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The following Ramsey-type question is one of the central problems in combinatorial geometry. Given a collection of certain geometric objects in the plane (e.g. segments, rectangles, convex sets, arcwise connected sets) of size n, what is the size of the largest subcollection in which either any two elements have a nonempty intersection, or any two elements are disjoint? We prove that there exists an absolute constant c>0 such that if C is a collection of n curves in the plane, then C contains at least nc elements that are pairwise intersecting, or n c elements that are pairwise disjoint. This resolves a problem raised by Alon, Pach, Pinchasi, Radoičić and Sharir, and Fox and Pach. Furthermore, as any geometric object can be arbitrarily closely approximated by a curve, this shows that the answer to the aforementioned question is at least nc for any collection of n geometric objects. [ABSTRACT FROM AUTHOR]

We consider the stochastic Navier–Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on three examples of a stochastic perturbation: an additive, a linear multiplicative and a nonlinear noise of cylindrical type, all driven by a Wiener process. In these settings, we develop a stochastic counterpart of the convex integration method introduced recently by Buckmaster and Vicol. This permits us to construct probabilistically strong and analytically weak solutions defined up to a suitable stopping time. In addition, these solutions fail to satisfy the corresponding energy inequality at a prescribed time with a prescribed probability. Then we introduce a general probabilistic construction used to extend the convex integration solutions beyond the stopping time and in particular to the whole time interval [0,∞). Finally, we show that their law is distinct from the law of solutions obtained by Galerkin approximation. In particular, non uniqueness in law holds on an arbitrary time interval[0,T], T > 0. [ABSTRACT FROM AUTHOR]