On randomly generated non-trivially intersecting hypergraphs.
On sets of integers containing k elements in arithmetic progression
On sumsets and spectral gaps
On the degree of regularity of generalized van der Waerden triples.
On the dimension of additive sets
We study the relations between several notions of dimension for an additive set, some of which are well-known and some of which are more recent, appearing for instance in work of Schoen and Shkredov. We obtain bounds for the ratios between these dimensions by improving an inequality of Lev and Yuster, and we show that these bounds are asymptotically sharp, using in particular the existence of large dissociated subsets of {0,1}ⁿ ⊂ ℤⁿ.
On the growth of a van der Waerden-like function.
On the locality of the Prüfer code.
On the monochromatic Schur triples type problem.
On the most weight vectors in a dimension binary code.
On the size of dissociated bases.
On the structure of sets with many k-term arithmetic progressions
On the total k-domination number of graphs
Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V, . Also the total k-domination number of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V, . The k-transversal number τₖ(H) of a hypergraph H is the minimum size of a subset S ⊆ V(H) such that |S ∩e | ≥ k for every edge e ∈ E(H). We know that for any graph...
On universal graphs for hom-properties
A graph property is any isomorphism closed class of simple graphs. For a simple finite graph H, let → H denote the class of all simple countable graphs that admit homomorphisms to H, such classes of graphs are called hom-properties. Given a graph property 𝓟, a graph G ∈ 𝓟 is universal in 𝓟 if each member of 𝓟 is isomorphic to an induced subgraph of G. In particular, we consider universal graphs in → H and we give a new proof of the existence of a universal graph in → H, for any finite graph...
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