Two color off-diagonal Rado-type numbers.
Two new extensions of the Hales-Jewett theorem.
Two results on a partial ordering of finite sequences
In the first part of the paper we are concerned about finite sequences (over arbitrary symbols) for which . The function measures the maximum length of finite sequences over symbols which contain no subsequence of the type . It follows from the result of Hart and Sharir that the containment is a (minimal) obstacle to . We show by means of a construction due to Sharir and Wiernik that there is another obstacle to the linear growth. In the second part of the paper we investigate whether...
Two variants of the size Ramsey number
Given a graph H and an integer r ≥ 2, let G → (H,r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let and define the Ramsey density as the infimum of m(G) over all graphs G such that G → (H,r). In the first part of this paper we show that when H is a complete graph Kₖ on k vertices, then , where R = R(k;r) is the classical Ramsey number. As a corollary we derive a new proof of the result credited to Chvatál...
Unification of zero-sum problems, subset sums and covers of .
Uniform edge distribution in hypergraphs is hereditary.
Universal minimal flows of automorphism groups
Universally image partition regularity.
Upper and lower bounds for a function related to Brown's Lemma.
Using Lovász local lemma in the space of random injections.
Van der Waerden's theorem and avoidability in words.
Zero-sum bipartite Ramsey numbers