Let G be a 2-connected graph of order n satisfying α(G) = a ≤ κ(G), where α(G) and κ(G) are the independence number and the connectivity of G, respectively, and let r(m,n) denote the Ramsey number. The well-known Chvátal-Erdös Theorem states that G has a hamiltonian cycle. In this paper, we extend this theorem, and prove that G has a 2-factor with a specified number of components if n is sufficiently large. More precisely, we prove that (1) if n ≥ k·r(a+4, a+1), then G has a 2-factor with k components,...

For any two graphs F1 and F2, the graph Ramsey number r(F1, F2) is the smallest positive integer N with the property that every graph on at least N vertices contains F1 or its complement contains F2 as a subgraph. In this paper, we consider the Ramsey numbers for theta-complete graphs. We determine r(θn,Km) for m = 2, 3, 4 and n > m. More specifically, we establish that r(θn,Km) = (n − 1)(m − 1) + 1 for m = 3, 4 and n > m

Bondy and Erdős [2] have conjectured that the Ramsey number for three cycles Cₖ of odd length has value r(Cₖ,Cₖ,Cₖ) = 4k-3. We give a proof that r(C₇,C₇,C₇) = 25 without using any computer support.

A 3-uniform hypergraph is called a minimum 3-tree, if for any 3-coloring of its vertex set there is a heterochromatic triple and the hypergraph has the minimum possible number of triples. There is a conjecture that the number of triples in such 3-tree is ⎡(n(n-2))/3⎤ for any number of vertices n. Here we give a proof of this conjecture for any n ≡ 0,1 mod 12.

The upper domination Ramsey number u(m,n) is the smallest integer p such that every 2-coloring of the edges of Kₚ with color red and blue, Γ(B) ≥ m or Γ(R) ≥ n, where B and R is the subgraph of Kₚ induced by blue and red edges, respectively; Γ(G) is the maximum cardinality of a minimal dominating set of a graph G. In this paper, we show that u(4,4) ≤ 15.

The focus of this article is on three of the author's open conjectures. The article itself surveys results relating to the conjectures and shows where the conjectures are known to hold.

We investigate the connections between Ramsey properties of Fraïssé classes and the universal minimal flow $M(G_{)}$ of the automorphism group $G$ of their Fraïssé limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class has finite Ramsey degree for embeddings, then this degree equals the size of $M(G_{)}$. We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if is a relational Ramsey class and $G$ is amenable, then $M(G_{)}$ admits a unique invariant...

Let T¹ₙ = (V,E₁) and T²ₙ = (V,E₂) be the trees on n vertices with $V=v₀,v₁,...,v_{n-1}$, $E₁=v₀v₁,...,v₀v_{n-3},v_{n-4}{v}_{n-2},v_{n-3}{v}_{n-1}$ and $E₂=v₀v₁,...,v₀v_{n-3},v_{n-3}{v}_{n-2},v_{n-3}{v}_{n-1}$. For p ≥ n ≥ 5 we obtain explicit formulas for ex(p;T¹ₙ) and ex(p;T²ₙ), where ex(p;L) denotes the maximal number of edges in a graph of order p not containing L as a subgraph. Let r(G₁,G₂) be the Ramsey number of the two graphs G₁ and G₂. We also obtain some explicit formulas for $r(Tₘ,T{ₙ}^i)$, where i ∈ 1,2 and Tₘ is a tree on m vertices with Δ(Tₘ) ≤ m - 3.

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 $m\left(G\right)=ma{x}_{F\subseteq G}\left|E\left(F\right)\right|/\left|V\left(F\right)\right|$ and define the Ramsey density $m_{inf}(H,r)$ 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 $m_{inf}(H,r)=(R-1)/2$, where R = R(k;r) is the classical Ramsey number. As a corollary we derive a new proof of the result credited to Chvatál...