Folkman Numbers Research Articles

On Some Generalized Vertex Folkman Numbers

For a graph G and integers $$a_i\ge 1$$ , the expression $$G \rightarrow (a_1,\ldots ,a_r)^v$$ means that for any r-coloring of the vertices of G there exists a monochromatic $$a_i$$ -clique in G for some color $$i \in \{1,\ldots ,r\}$$ . The vertex Folkman numbers are defined as $$F_v(a_1,\ldots ,a_r;H) = \min \{|V(G)| : G$$ is H-free and $$G \rightarrow (a_1,\ldots ,a_r)^v\}$$ , where H is a graph. Such vertex Folkman numbers have been extensively studied for $$H=K_s$$ with $$s>\max \{a_i\}_{1\le i \le r}$$ . If $$a_i=a$$ for all i, then we use notation $$F_v(a^r;H)=F_v(a_1,\ldots ,a_r;H)$$ . Let $$J_k$$ be the complete graph $$K_k$$ missing one edge, i.e. $$J_k=K_k-e$$ . In this work we focus on vertex Folkman numbers with $$H=J_k$$ , in particular for $$k=4$$ and $$a_i\le 3$$ . A result by Nešetřil and Rödl from 1976 implies that $$F_v(3^r;J_4)$$ is well defined for any $$r\ge 2$$ . We present a new and more direct proof of this fact. The simplest but already intriguing case is that of $$F_v(3,3;J_4)$$ , for which we establish the upper bound of 135 by using the $$J_4$$ -free process. We obtain the exact values and bounds for a few other small cases of $$F_v(a_1,\ldots ,a_r;J_4)$$ when $$a_i \le 3$$ for all $$1 \le i \le r$$ , including $$F_v(2,3;J_4)=14$$ , $$F_v(2^4;J_4)=15$$ , and $$22 \le F_v(2^5;J_4) \le 25$$ . Note that $$F_v(2^r;J_4)$$ is the smallest number of vertices in any $$J_4$$ -free graph with chromatic number $$r+1$$ . Most of the results were obtained with the help of computations, but some of the upper bound graphs we found are interesting by themselves.

Chromatic Vertex Folkman Numbers

For graph $G$ and integers $a_1 \ge \cdots \ge a_r \ge 2$, we write $G \rightarrow (a_1 ,\cdots ,a_r)^v$ if and only if for every $r$-coloring of the vertex set $V(G)$ there exists a monochromatic $K_{a_i}$ in $G$ for some color $i \in \{1, \cdots, r\}$. The vertex Folkman number $F_v(a_1 ,\cdots ,a_r; s)$ is defined as the smallest integer $n$ for which there exists a $K_s$-free graph $G$ of order $n$ such that $G \rightarrow (a_1 ,\cdots ,a_r)^v$. It is well known that if $G \rightarrow (a_1 ,\cdots ,a_r)^v$ then $\chi(G) \geq m$, where $m = 1+ \sum_{i=1}^r (a_i - 1)$. In this paper we study such Folkman graphs $G$ with chromatic number $\chi(G)=m$, which leads to a new concept of chromatic Folkman numbers. We prove constructively some existential results, among others that for all $r,s \ge 2$ there exist $K_{s+1}$-free graphs $G$ such that $G \rightarrow (s,\cdots_r,s)^v$ and $G$ has the smallest possible chromatic number $r(s-1)+1$ with respect to this property. Among others we conjecture that for every $s \ge 2$ there exists a $K_{s+1}$-free graph $G$ on $F_v(s,s;s+1)$ vertices with $\chi(G)=2s-1$ and $G\rightarrow (s,s)^v$.

On some edge Folkman numbers, small and large

On some edge Folkman numbers, small and large

A note on upper bounds for some generalized 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 $F_e(K_3,K_4-e; K_5) \leq 27$ and $F_e(K_4-e,K_4-e; K_5) \leq 51$. The latter bound follows from a construction of a $K_5$-free graph on 51 vertices, for which every coloring of its edges with two colors contains a monochromatic $K_4-e$.

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.

Vertex Folkman Numbers and the Minimum Degree of Minimal Ramsey Graphs

We investigate the smallest possible minimum degree of $r$-color minimal Ramsey graphs for the $k$-clique. In particular, we obtain a bound of the form $O(k^2\log^2 k\big)$, which is tight up to a $(\log^2 k)$-factor whenever the number $r\geq2$ of colors is fixed. This extends the work of Burr, Erdos, and Lovasz, who determined this extremal value for two colors and any clique size, and complements that of Fox, Grinshpun, Liebenau, Person, and Szabo, who gave essentially tight bounds when the order $k$ of the clique is fixed. As a side product our result also yields an improved upper bound on the vertex Folkman number $F(r,k, k+1)$ of the $k$-clique. The proof relies on a reformulation of the corresponding extremal function by Fox et al. and combines and refines methods used by Dudek, Eaton, and Rodl.

Ramsey properties of random graphs and Folkman numbers

For two graphs, $G$ and $F$, and an integer $r\ge2$ we write $G\rightarrow (F)_r$ if every $r$-coloring of the edges of $G$ results in a monochromatic copy of $F$. In 1995, the first two authors established a threshold edge probability for the Ramsey property $G(n,p)\to (F)_r$, where $G(n,p)$ is a random graph obtained by including each edge of the complete graph on $n$ vertices, independently, with probability $p$. The original proof was based on the regularity lemma of Szemer\'edi and this led to tower-type dependencies between the involved parameters. Here, for $r=2$, we provide a self-contained proof of a quantitative version of the Ramsey threshold theorem with only double exponential dependencies between the constants. As a corollary we obtain a double exponential upper bound on the 2-color Folkman numbers. By a different proof technique, a similar result was obtained independently by Conlon and Gowers.

An exponential-type upper bound for Folkman numbers

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 order $k$. The existence (finiteness) of Folkman numbers was established by Folkman (1970) for $r=2$ and by Ne\v{s}et\v{r}il and R\"odl (1976) for arbitrary $r$, but these proofs led to very weak upper bounds on $f(k;r)$. Recently, Conlon and Gowers and independently the authors obtained a doubly exponential bound on $f(k;2)$. Here, we establish a further improvement by showing an upper bound on $f(k;r)$ which is exponential in a polynomial function of $k$ and $r$. This is comparable to the known lower bound $2^{\Omega(rk)}$. Our proof relies on a recent result of Saxton and Thomason (2015) (or, alternatively, on a recent result of Balogh, Morris, and Samotij (2015)) from which we deduce a quantitative version of Ramsey's theorem in random graphs.

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.

Some remarks on vertex Folkman numbers for hypergraphs

Let F(r,G) be the least order of H such that the clique numbers of H and G are equal and any r-coloring of the vertices of H yields a monochromatic and induced copy of G. The problem of bounding of F(r,G) was studied by several authors and it is well understood. In this note, we extend those results to uniform hypergraphs.

On induced Folkman numbers

AbstractIn 1970, Folkman proved that for any graph G there exists a graph H with the same clique number as G. In addition, any r ‐coloring of the vertices of H yields a monochromatic copy of G. For a given graph G and a number of colors r let f(G, r) be the order of the smallest graph H with the above properties. In this paper, we give a relatively small upper bound on f(G, r) as a function of the order of G and its clique number. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 40, 493–500, 2012

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

Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers 2 ≤ s 0 and sufficiently large integers 1 ≪ k ≪ s, Ω(n 1/2−ɛ ) ≤ f s,s+k (n) ≤ O(n 1/2+ɛ . In addition, we also discuss some connections between the function f s,t and vertex Folkman numbers.

Some recent results on Ramsey-type numbers

In this paper we survey the authors’ recent results on quantitative extensions of Ramsey theory. In particular, we discuss our recent results on Folkman numbers, induced bipartite Ramsey graphs, and explicit constructions of Ramsey graphs.

An almost quadratic bound on vertex Folkman numbers

The vertex Folkman number F ( r , n , m ) , n < m , is the smallest integer t such that there exists a K m -free graph of order t with the property that every r-coloring of its vertices yields a monochromatic copy of K n . The problem of bounding the Folkman numbers has been studied by several authors. However, in the most restrictive case, when m = n + 1 , no polynomial bound has been known for such numbers. In this paper we show that the vertex Folkman numbers F ( r , n , n + 1 ) are bounded from above by O ( n 2 log 4 n ) . Furthermore, for any fixed r and any small ε > 0 we derive the linear upper bound when the cliques bigger than ( 2 + ε ) n are forbidden.

Finding Folkman Numbers via MAX CUT Problem

Abstract In this note we report on our recent work, still in progress, regarding Folkman numbers. Let f ( 2 , 3 , 4 ) denote the smallest integer n such that there exists a K 4 -free graph of order n having that property that any 2-coloring of its edges yields at least one monochromatic triangle. It is well-known that such a number must exist [J. Folkman, Graphs with monochromatic complete subgraphs in every edge coloring, SIAM J. Appl. Math., 18 (1970), 19–24, J. Nesetřil and V. Rodl, The Ramsey property for graphs with forbidden complete subgraphs, J. Comb. Theory Ser. B, 20, (1976), 243–249]. For almost twenty years the best known upper bound, given by Spencer, was f ( 2 , 3 , 4 ) 3 ⋅ 10 9 [J. Spencer, Three hundred million points suffice, J. Comb. Theory Ser. A, 49 (1988), 210–217. Also see erratum by M. Hovey in Vol. 50, p. 323]. Recently, the authors and Lu showed that f ( 2 , 3 , 4 ) 130000 [A. Dudek and V. Rodl, On the Folkman number f(2, 3, 4), submitted] and f ( 2 , 3 , 4 ) 10000 [L. Lu, Explicit construction of small Folkman graphs, submitted]. However, it is commonly believed that, in fact, f ( 2 , 3 , 4 ) 100 . All previous bounds are based on an idea of Goodman [A.W. Goodman, On sets of acquaintances and strangers at any party, Amer. Math. Mon., 66 (1959), 778–783]. It seems that such methods will not yield substantial further improvement. In this note we will generalize this idea by giving a necessary and sufficient condition for a graph G to yield a monochromatic triangle for every edge coloring. In particular, for any graph G we construct a graph H such that G is Folkman if and only if the value of the maximum cut of H is less than twice the number of triangles in G. We believe this technique may be used to find a new upper bound on f ( 2 , 3 , 4 ) .

A multiplicative inequality for vertex Folkman numbers

Let G be a graph and a 1 , … , a r be positive integers. The symbol G → ( a 1 , … , a r ) denotes that in every r-coloring of the vertex set V ( G ) there exists a monochromatic a i -clique of color i for some i ∈ { 1 , … , r } . The vertex Folkman numbers F ( a 1 , … , a r ; q ) = min { | V ( G ) | : G → ( a 1 , … , a r ) and K q ⊈ G } are considered. Let a i , b i , c i , i ∈ { 1 , … , r } , s, t be positive integers and c i = a i b i , 1 ⩽ a i ⩽ s , 1 ⩽ b i ⩽ t . Then we prove that F ( c 1 , c 2 , … , c r ; st + 1 ) ⩽ F ( a 1 , a 2 , … , a r ; s + 1 ) F ( b 1 , b 2 , … , b r ; t + 1 ) .

New Upper Bound for a Class of Vertex Folkman Numbers

Let $a_1, \ldots, a_r$ be positive integers, $m=\sum_{i=1}^{r} (a_{i}-1)+1$ and $p= \max \{a_1, \ldots, a_r\}$. For a graph $G$ the symbol $G\rightarrow \{a_1, \ldots, a_r\}$ denotes that in every $r$-coloring of the vertices of $G$ there exists a monochromatic $a_i$-clique of color $i$ for some $i=1, \ldots, r$. The vertex Folkman numbers $F(a_1,\dots,a_r;m-1)=\min \{| V(G) | : G\rightarrow (a_1 \ldots a_r)$ and $K_{m-1} \not \subseteq G \}$ are considered. We prove that $F(a_1, \ldots, a_r; m-1) \leq m+3p$, $p \geq 3$. This inequality improves the bound for these numbers obtained by Łuczak, Ruciński and Urbański (2001).

On the triangle vertex Folkman numbers

For a graph G the symbol G→( 3,…,3 r ) means that in every r-colouring of the vertices of G there exists a monochromatic triangle. The triangle vertex Folkman numbers F r(3)= min{|V(G)|:G→( 3,…,3 r ) and cl( G)<2 r} are considered. We prove that F r (3)=2 r+7, r⩾3.

Computation of the Vertex Folkman Numbers $F(2,2,2,4;6)$ and $F(2,3,4;6)$

In this note we show that the exact value of the vertex Folkman numbers $F(2,2,2,4;6)$ and $F(2,3,4;6)$ is 14.

Computation of the Folkman numberFe(3, 3; 5)

With the help of computer algorithms, we improve the lower bound on the edge Folkman number Fe(3, 3; 5) and vertex Folkman number Fv(3, 3; 4), and, thus, show that the exact values of these numbers are 15 and 14, respectively. We also present computer enumeration of all critical graphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 41–49, 1999