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Displaying 61 –
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We study the thresholds for the emergence of various properties in random subgraphs of (ℕ, <). In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability is made. The main tools are a topological version of Ramsey theory, exchangeability theory and elementary ergodic theory.
In this note, we introduce the notion of -Ramsey classes of graphs and we reveal connections to intersection dimensions of graphs.
Let H be a fixed finite graph and let → H be a hom-property, i.e. the set of all graphs admitting a homomorphism into H. We extend the definition of → H to include certain infinite graphs H and then describe the minimal reducible bounds for → H in the lattice of additive hereditary properties and in the lattice of hereditary properties.
For given graphs G₁,G₂,...,Gₖ, k ≥ 2, the multicolor Ramsey number R(G₁,G₂,...,Gₖ) is the smallest integer n such that if we arbitrarily color the edges of the complete graph on n vertices with k colors, then it is always a monochromatic copy of some , for 1 ≤ i ≤ k. We give a lower bound for k-color Ramsey number R(Cₘ,Cₘ,...,Cₘ), where m ≥ 8 is even and Cₘ is the cycle on m vertices. In addition, we provide exact values for Ramsey numbers R(P₃,Cₘ,Cₚ), where P₃ is the path on 3 vertices, and several...
We give the multicolor Ramsey number for some graphs with a path or a cycle in the given sequence, generalizing a results of Faudree and Schelp [4], and Dzido, Kubale and Piwakowski [2,3].
2000 Mathematics Subject Classification: 05C55.For a given graph G let V(G) and E(G) denote the vertex and the edge set of G respevtively.
The symbol G e → (a1, …, ar)
means that in every r-coloring of E(G) there exists a monochromatic ai-clique of color i
for some i ∈ {1,…,r}. The edge Folkman numbers are defined by the equality
Fe(a1, …, ar; q) = min{|V(G)| : G e → (a1, …, ar; q) and cl(G) < q}.
In this paper we prove a new upper bound on the edge Folkman number Fe(3,5;13), namely
Fe(3,5;13)...
A set X of vertices of a graph G is said to be 1-dependent if the subgraph of G induced by X has maximum degree one. The 1-dependent Ramsey number t₁(l,m) is the smallest integer n such that for any 2-edge colouring (R,B) of Kₙ, the spanning subgraph B of Kₙ has a 1-dependent set of size l or the subgraph R has a 1-dependent set of size m. The 2-edge colouring (R,B) is a t₁(l,m) Ramsey colouring of Kₙ if B (R, respectively) does not contain a 1-dependent set of size l (m, respectively); in this...
Let a1 , . . . , ar, be positive integers, i=1 ... r, m = ∑(ai − 1) + 1 and
p = max{a1 , . . . , ar }. For a graph G the symbol G → (a1 , . . . , ar ) means
that in every r-coloring of the vertices of G there exists a monochromatic
ai -clique of color i for some i ∈ {1, . . . , r}. In this paper we consider the
vertex Folkman numbers
F (a1 , . . . , ar ; m − 1) = min |V (G)| : G → (a1 , . . . , ar ) and Km−1 ⊂ G}
We prove that F (a1 , . . . , ar ; m − 1) = m + 6, if p = 3 and m ≧ 6 (Theorem
3)...
We discuss dual Ramsey statements for several classes of finite relational structures (such as finite linearly ordered graphs, finite linearly ordered metric spaces and finite posets with a linear extension) and conclude the paper with another rendering of the Nešetřil-Rödl Theorem for relational structures. Instead of embeddings which are crucial for ``direct'' Ramsey results, for each class of structures under consideration we propose a special class of quotient maps and prove a dual Ramsey theorem...
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...
Currently displaying 61 –
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141