Now, in analogy to Ramsay's theorem, one might consider the following problem. Suppose that, for some u > 0, there is associated with each k-tuple X = {x l , • • • , xk } of elements of an infinite set S a measurable set F(X) of [0, 1] such that m(F(X)) > u . Does there always exist an infinite subset S' of S such that the sets F(X) corresponding to the k-tuples X of S' have a nonempty intersec...

We prove a version of B ezout's theorem for non-commutative analogues of the projective spaces P n. 0. Introduction Throughout we work over an algebraically closed eld k. We establish a version of B ezout's Theorem for non-commutative projective spaces, quantum P n s for short. If Y is a quantum P n then the alternating sum of the dimension of the Ext-groups gives a bilinear form : K 0 (Y) K 0 ...

In this paper, we prove a general fixed point theorem in $textrm{S}$-metric spaces for maps satisfying an implicit relation on complete metric spaces. As applications, we get many analogues of fixed point theorems in metric spaces for $textrm{S}$-metric spaces.

This short note gives an alternative proof of the M-convex intersection theorem, which is one of the central results in discrete convex analysis. This note is intended to provide a direct simpler proof accessible to nonexperts. 1 M-Convex Intersection Theorem The M-convex intersection theorem [3, Theorem 8.17] reads as follows, where V is a nonempty finite set, and Z and R are the sets of integ...

We establish several geometric extensions of the Lipton-Tarjan separator theorem for planar graphs. For instance, we show that any collection C of Jordan curves in the plane with a total of m crossings has a partition into three parts C = S ∪C1 ∪C2 such that |S| = O( √ m), max{|C1|, |C2|} ≤ 2 3 |C|, and no element of C1 has a point in common with any element of C2. These results are used to obt...

A family F of sets has property B(s) if there exists a set S whose intersection with each set in F is non-empty but contains fewer than s elements . P . Erdős has asked whether there exists an absolute constant c such that every projective plane has property B(c) . In this paper, the authors, as a partial answer to this question, obtain the result that for n sufficiently large, every projective...

The dimension of a poset (X, P), denoted dim (A", P), is the minimum number of linear extensions of P whose intersection is P. It follows from Dilworth's decomposition theorem that dim (X, P)& width (X, P). Hiraguchi showed that dim(X, P)s \X\/Z In this paper, A denotes an antichain of (A", P) and E the set of maximal elements. We then prove that dim {X, P) s |X A\; dim(X, P) < 1 + width (X E);...