On the triangle vertex Folkman numbers

Nenov, N.

doi:10.1016/s0012-365x(03)00134-1

Cited by 7 publications

(6 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It follows from (4) that in every 2-coloring of V (Q), in which there are no 4-cliques in none of the two colors then there are 3-cliques in the both colors. Therefore the inequalities in (10) are in fact equalities, which contradicts (7). Thus (8) is proved.…”

Section: Proof Of the Theoremmentioning

confidence: 83%

New Upper Bound for a Class of Vertex Folkman Numbers

Kolev¹,

Nenov²

2006

Electron. J. Combin.

View full text Add to dashboard Cite

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 rcoloring 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; 8), namely Fe(3, 5; 8) ≤ 21 This improves the bound Fe(3, 5; 8) ≤ 24, proved in [5]

show abstract

Section: Proof Of the Theoremmentioning

confidence: 83%

New Upper Bound for a Class of Vertex Folkman Numbers

Kolev¹,

Nenov²

2006

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…The cases k = 1 and k = 2 of Theorem 1.1 were proved in [18]. It was also proved in [18] that K r−5 + C 5 + C 5 , r ≥ 5 is the only minimal graph in H v (2 r ; r) (see also [23]).…”

Section: By (11) (12)mentioning

confidence: 89%

On the Vertex Folkman Numbers $F_v(2,...,2;q)$

Nenov

2009

Preprint

View full text Add to dashboard Cite

For a graph G the symbol G v → (a 1 , . . . , a r ) means that in every r-coloring of the vertices of G for some i ∈ {1, . . . , r} there exists a monochromatic a i -clique of color i. The vertex Folkman numbers. . , a r ) and K q G} are considered. In this paper we shall compute the Folkman numbers F v (2, . . . , 2 r ; r− k + 1) when k ≤ 12 and r is sufficiently large. We prove also new bounds for some vertex and edge Folkman numbers.

show abstract

“…This function was examined by several researchers, see, e.g., [4,8,10,[13][14][15]. Folkman [8] showed that F (r, s, t) is well-defined by giving an enormous upper bound.…”

Section: Vertex Folkman Numbersmentioning

confidence: 99%

On K s -free subgraphs in K s+k -free graphs and vertex Folkman numbers

Dudek

Rödl

2011

Combinatorica

View full text Add to dashboard Cite

Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers 2 ≤ s < t let fs,t(n) = min˘max{|S| : S ⊆ V (H) and H [S] contains no Ks}¯, where the minimum is taken over all Kt-free graphs H of order n. This function attracted a considerable amount of attention but despite that, the gap between the lower and upper bounds is still fairly wide. For example, when t = s+1, the best bounds have been of the form Ω`n 1 2 +o(1)´≤ fs,s+1(n) ≤ O(n 1− (s) ), where (s) tends to zero as s tends to infinity. In this paper we improve the upper bound by showing that fs,s+1(n) ≤ O`n 2 3´. Moreover, we show that for every ε > 0 and sufficiently large integers 1 k s, Ω`n 1 2 −ε´≤ f s,s+k (n) ≤ O`n 1 2 +ε´. In addition, we also discuss some connections between the function fs,t and vertex Folkman numbers.

show abstract

On the triangle vertex Folkman numbers

Cited by 7 publications

References 7 publications

New Upper Bound for a Class of Vertex Folkman Numbers

New Upper Bound for a Class of Vertex Folkman Numbers

On the Vertex Folkman Numbers $F_v(2,...,2;q)$

On K s -free subgraphs in K s+k -free graphs and vertex Folkman numbers

Contact Info

Product

Resources

About