On a classification of denumerable order types and an application to the partition calculus
On a Rado type problem for homogeneous second order linear recurrences.
On a variant of van der Waerden's theorem.
On Hales-Jewett's theorem.
On monochromatic sets of integers whose diameters form a monotone sequence.
On monochromatic solutions of equations in groups.
On Monochromatic Subgraphs of Edge-Colored Complete Graphs
In a red-blue coloring of a nonempty graph, every edge is colored red or blue. If the resulting edge-colored graph contains a nonempty subgraph G without isolated vertices every edge of which is colored the same, then G is said to be monochromatic. For two nonempty graphs G and H without isolated vertices, the mono- chromatic Ramsey number mr(G,H) of G and H is the minimum integer n such that every red-blue coloring of Kn results in a monochromatic G or a monochromatic H. Thus, the standard Ramsey...
On path-quasar Ramsey numbers
Let G1 and G2 be two given graphs. The Ramsey number R(G1,G2) is the least integer r such that for every graph G on r vertices, either G contains a G1 or Ḡ contains a G2. Parsons gave a recursive formula to determine the values of R(Pn,K1,m), where Pn is a path on n vertices and K1,m is a star on m+1 vertices. In this note, we study the Ramsey numbers R(Pn,K1,m), where Pn is a linear forest on m vertices. We determine the exact values of R(Pn,K1∨Fm) for the cases m ≤ n and m ≥ 2n, and for the case...
On rainbow arithmetic progressions.
On rainbow solutions to an equation with a quadratic term.
On Ramsey -minimal graphs
For graphs F, G and H, we write F → (G,H) to mean that any red-blue coloring of the edges of F contains a red copy of G or a blue copy of H. The graph F is Ramsey (G,H)-minimal if F → (G,H) but F* ↛ (G,H) for any proper subgraph F* ⊂ F. We present an infinite family of Ramsey -minimal graphs of any diameter ≥ 4.
On the degree of regularity of generalized van der Waerden triples.
On the growth of a van der Waerden-like function.
On the monochromatic Schur triples type problem.
One More Turán Number and Ramsey Number for the Loose 3-Uniform Path of Length Three
Let P denote a 3-uniform hypergraph consisting of 7 vertices a, b, c, d, e, f, g and 3 edges {a, b, c}, {c, d, e}, and {e, f, g}. It is known that the r-color Ramsey number for P is R(P; r) = r + 6 for r ≤ 9. The proof of this result relies on a careful analysis of the Turán numbers for P. In this paper, we refine this analysis further and compute the fifth order Turán number for P, for all n. Using this number for n = 16, we confirm the formula R(P; 10) = 16.
On-line Ramsey theory.
Ordinal types in Ramsey theory and well-partial-ordering theory