Related papers: New Upper Bound for the Edge Folkman Number Fe(3,5…
In this paper we discuss a class of combinatorial constants in Ramsey theory- edge Folkman numbers. We give an upper bound on one of them- the number F_e(3,3,3;13).
The set of the graphs which do not contain the complete graph on $q$ vertices $K_q$ and have the property that in every coloring of their edges in two colors there exist a monochromatic triangle is denoted by $\mathcal{H}_e(3, 3; q)$. The…
In this paper we prove a new upper bouhd on an edge Folkman number. In a previous paper we have proved that this bound is exact.
We give some bounds on edge Folkman numbers.
We present some new constructive upper bounds based on product graphs for generalized vertex Folkman numbers. They lead to new upper bounds for some special cases of generalized edge Folkman numbers, including $F_e(K_3,K_4-e; K_5) \leq 27$…
For a graph $G$ the expression $G \overset{v}{\rightarrow} (a_1, ..., a_s)$ means that for every coloring of the vertices of $G$ in $s$ colors there exists $i \in \{1, ..., s\}$ such that there is a monochromatic $a_i$-clique of color $i$.…
We improve the previuosly known bound for some vertex Folkman numbers.
For a graph $G$ the expression $G \overset{v}{\rightarrow} (a_1, ..., a_s)$ means that for every $s$-coloring of the vertices of $G$ there exists $i \in \{1, ..., s\}$ such that there is a monochromatic $a_i$-clique of color $i$. The vertex…
The graph $G$ is called a $(3, 3)$-Ramsey graph if in every coloring of the edges of $G$ in two colors there is a monochromatic triangle. The minimum number of vertices of the $(3, 3)$-Ramsey graphs without 4-cliques is denoted by $F_e(3,…
In the paper we give a lower bound for the number of vertices of a given graph using its chromatic number. We find the graphs for which this bound is exact. The results are applied in the theory of Foklman numbers.
For a graph $G$ the symbol $G\tov(a_1,...,a_r)$ means that in every $r$-coloring of the vertices of $G$ for some $i\in\{1,...,r\}$ there exists a monochromatic $a_i$-clique of color $i$. The vertex Folkman numbers \[…
This first extracted report contains all lower and upper bounds for e-numbers $e(3,k;n)$, for $n \leq 43$, that I know. All but 24 of them are known (exactly).Very little of the proofs is given. A few consequences for upper classical Ramsey…
For given integers $k$ and $r$, the Folkman number $f(k;r)$ is the smallest number of vertices in a graph $G$ which contains no clique on $k+1$ vertices, yet for every partition of its edges into $r$ parts, some part contains a clique of…
Folkman's Theorem asserts that for each $k \in \mathbb{N}$, there exists a natural number $n = F(k)$ such that whenever the elements of $[n]$ are two-coloured, there exists a set $A \subset [n]$ of size $k$ with the property that all the…
Recent work has established that, for every positive integer $k$, every $n$-node graph has a $(2k-1)$-spanner on $O(f^{1-1/k} n^{1+1/k})$ edges that is resilient to $f$ edge or vertex faults. For vertex faults, this bound is tight. However,…
For a graph $G$ the expression $G \overset{v}{\rightarrow} (a_1, ..., a_s)$ means that for any $s$-coloring of the vertices of $G$ there exists $i \in \{1, ..., s\}$ such that there is a monochromatic $a_i$-clique of color $i$. The vertex…
We prove an exponential upper bound for the number $f(m,n)$ of all maximal triangulations of the $m\times n$ grid: \[ f(m,n) < 2^{3mn}. \] In particular, this improves a result of S. Yu. Orevkov (1999).
A fan $F_n$ is a graph consisting of $n$ triangles, all having precisely one common vertex. Currently, the best known bounds for the Ramsey number $R(F_n)$ are $9n/2-5 \leq R(F_n) \leq 11n/2+6$, obtained by Chen, Yu and Zhao. We improve the…
A graph drawn in the plane with n vertices is k-fan-crossing free for k > 1 if there are no k+1 edges $g,e_1,...e_k$, such that $e_1,e_2,...e_k$ have a common endpoint and $g$ crosses all $e_i$. We prove a tight bound of 4n-8 on the maximum…
In 1975, P. Erd\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+32t-1$$ for…