A Folkman Linear Family

For graphs $F$ and $G$, let $F\to (G,G)$ signify that any red/blue edge coloring of $F$ contains a monochromatic $G$. Define Folkman number $f(G;p)$ to be the smallest order of a graph $F$ such that $F\to (G,G)$ and $\omega(F) \le p$. It is shown that $f(G;p)\le cn$ for graphs $G$ of order $n$ with $\Delta(G)\le \Delta$, where $\Delta\ge 3$, $c=c(\Delta)$ and $p=p(\Delta)$ are positive constants.Comment: 11 page

Combinatorial theorems relative to a random set

We describe recent advances in the study of random analogues of combinatorial theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201

Chromatic Vertex Folkman Numbers

For graph G and integers a1 \u3e · · · \u3e ar \u3e 2, we write G → (a1, · · · , ar) v if and only if for every r-coloring of the vertex set V (G) there exists a monochromatic Kai in G for some color i ∈ {1, · · · , r}. The vertex Folkman number Fv(a1, · · · , ar; s) is defined as the smallest integer n for which there exists a Ks-free graph G of order n such that G → (a1, · · · , ar) v . It is well known that if G → (a1, · · · , ar) v then χ(G) \u3e m, where m = 1+Pr i=1(ai−1). In this paper we study such Folkman graphs G with chromatic number χ(G) = m, which leads to a new concept of chromatic Folkman numbers. We prove constructively some existential results, among others that for all r, s \u3e 2 there exist Ks+1-free graphs G such that G → (s, · · ·r , s) v and G has the smallest possible chromatic number r(s − 1) + 1 with respect to this property. Among others we conjecture that for every s \u3e 2 there exists a Ks+1-free graph G on Fv(s, s; s + 1) vertices with χ(G) = 2s − 1 and G → (s, s) v

On Some Edge Folkman Numbers Small and Large

Edge Folkman numbers Fe(G1, G2; k) can be viewed as a generalization of more commonly studied Ramsey numbers. Fe(G1, G2; k) is defined as the smallest order of any Kk -free graph F such that any red-blue coloring of the edges of F contains either a red G1 or a blue G2. In this note, first we discuss edge Folkman numbers involving graphs Js = Ks − e, including the results Fe(J3, Kn; n + 1) = 2n − 1, Fe(J3, Jn; n) = 2n − 1, and Fe(J3, Jn; n + 1) = 2n − 3. Our modification of computational methods used previously in the study of classical Folkman numbers is applied to obtain upper bounds on Fe(J4, J4; k) for all k \u3e 4

On the Nonexistence of Some Generalized Folkman Numbers

For an undirected simple graph $G$, we write $G \rightarrow (H_1, H_2)^v$ if and only if for every red-blue coloring of its vertices there exists a red $H_1$ or a blue $H_2$. The generalized vertex Folkman number $F_v(H_1, H_2; H)$ is defined as the smallest integer $n$ for which there exists an $H$-free graph $G$ of order $n$ such that $G \rightarrow (H_1, H_2)^v$. The generalized edge Folkman numbers $F_e(H_1, H_2; H)$ are defined similarly, when colorings of the edges are considered. We show that $F_e(K_{k+1},K_{k+1};K_{k+2}-e)$ and $F_v(K_k,K_k;K_{k+1}-e)$ are well defined for $k \geq 3$. We prove the nonexistence of $F_e(K_3,K_3;H)$ for some $H$, in particular for $H=B_3$, where $B_k$ is the book graph of $k$ triangular pages, and for $H=K_1+P_4$. We pose three problems on generalized Folkman numbers, including the existence question of edge Folkman numbers $F_e(K_3, K_3; B_4)$, $F_e(K_3, K_3; K_1+C_4)$ and $F_e(K_3, K_3; \overline{P_2 \cup P_3} )$. Our results lead to some general inequalities involving two-color and multicolor Folkman numbers