It is shown that if a graph ofv vertices can be drawn in the plane so that every edge crosses at mostk>0 others, then its number of edges cannot exceed 4.108√kv, and a better bound is established, (k+3)(v−2), which is tight fork=1 and 2.Expand

We summarize important recent advances in quantum metrology, in connection to experiments in cold gases, trapped cold atoms and photons. First we review simple metrological setups, such as quantum… Expand

Let ES(n) denote the least integer such that among any ES(n) points in general position in the plane there are always n in convex position. In 1935, P. Erdős and G. Szekeres showed that ES(n) exists… Expand

If a graph can be drawn in the plane so that every edge crosses at most three others, then its number of edges cannot exceed 5.5( v-2); and the crossing number of any graph is at least $\frac73e-\frac{25}3(v-2).$ Both bounds are tight up to an additive constant.Expand

It is shown that the maximum number of edges of a simple topological graph with n vertices and no k pairwise disjoint edges is O(nlog4k−8n) edges.Expand

This paper improves the bounds of Erdős, Lovász, et al. on the number of halving hyperplanes in higher dimensions by constructing a set of n points in the plane with ne-Omega k -sets.Expand

It is shown that G can be redrawn in such a way that the x-coordinates of the vertices remain unchanged and the edges become non-crossing straight-line segments.Expand

The complete set of generalized spin squeezing inequalities are determined, which can be used for the experimental detection of entanglement in a system of spin-1/2 particles in which the spins cannot be individually addressed.Expand

Abstract We prove that for every $k\,>\,1$ , there exist $k$ -fold coverings of the plane (i) with strips, (ii) with axis-parallel rectangles, and (iii) with homothets of any fixed concave… Expand