A note on order-type homogeneous point sets

Let OT_d(n) be the smallest integer N such that every N-element point sequence in R^d in general position contains an order-type homogeneous subset of size n, where a set is order-type homogeneous if all (d+1)-tuples from this set have the same orientation. It is known that a point sequence in R^d that is order-type homogeneous forms the vertex set of a convex polytope that is combinatorially equivalent to a cyclic polytope in R^d. Two famous theorems of Erdos and Szekeres from 1935 imply that OT_1(n) = Theta(n^2) and OT_2(n) = 2^(Theta(n)). For d \geq 3, we give new bounds for OT_d(n). In particular: 1. We show that OT_3(n) = 2^(2^(Theta(n))), answering a question of Eli\'a\v{s} and Matou\v{s}ek. 2. For d \geq 4, we show that OT_d(n) is bounded above by an exponential tower of height d with O(n) in the topmost exponent

On Quadruple Random Fixed Point Theorems in Partially Ordered Metric Spaces

Bhaskar and Lakshmikantham in [15] introduced the concept of coupled fixed point of a mapping and investigated the existence and uniquencess of a coupled fixed point theorem in partially ordered complete metric space. Lakshmikantham and Ciric [16] defined  mixed g-monotone property and coincidence point in partially ordered metric space. V. Berinde and M. Borcut[18] introduced the concept of triple fixed point and proved some related theorems. Following this trand,  Karapinar[19] introduced the nation of quadruple fixed point. The object of this note is to prove quadruple random fixed point theorem in partially ordered metric spaces

Coupled Fixed Point Theorem in Partially Ordered Metric Spaces

The present paper deals with some Coupled fixed point theorem for mapping having mixed monotone property in Partially Ordered Metric space. AMS Subject Classification: 47H10, 54H25. Keywords: fixed point, mixed monotone property,coupled fixed point