We are going to answer some open questions in the theory of hyperconvex metric spaces. We prove that in complete R-trees hyperconvex hulls are uniquely determined. Next we show that hyperconvexity of subsets of normed spaces implies their convexity if and only if the space under consideration is strictly convex. Moreover, we prove a Krein-Milman type theorem for Rtrees. Finally, we discuss a general construction of certain complete metric spaces. We analyse its particular cases to investigate hyperconvexity via measures of noncompactness.

The central sets theorem originally proven by H. Furstenburg is a powerful result which is applicable to derive many combinatorial conclusions. Furstenburg's original theorem applied to N and finitely many sequences in Z. Some strengthenings of this theorem have been derived first by V. Bergelson and N. Hindman in 1990. Later in 2008, D. De, N. Hindman, and D. Strauss proved a stronger version of the central sets theorem for arbitrary semigroups S which applied to all sequences in S. We provide here a generalization of the stronger version and some applications of this new generalization.

For a, r ∈ N, the set of positive integers, define F SP 2 (a, r) respectively SP 2 (a, r) to be the first n ∈ N, if such exists, such that whenever {1, 2, . . . , n} is r-colored, there exist x and y with a ≤ x < y such that {x, y, x + y, xy} is monochromatic (respectively {x + y, xy} is monochromatic). If no such n exists, the number is defined to be infinite. It is an old result of R. Graham that SP 2 (a, 2) is finite for all a. With that exception, the only cases (with r > 1) for which F SP 2 (a, r) or SP 2 (a, r) are known to be finite are those for which explicit values have been computed. In this paper, we provide exact values of F SP 2 (a, 2) for a ≤ 5 (of which F SP 2 (1, 2) and F SP 2 (2, 2) were previously known). We provide exact values of SP 2 (a, 3) for a ≤ 9 and exact values of SP 2 (a, 2) for a ≤ 105. We also compute upper and lower bounds for SP 2 (a, 2).