On the independence number of $(3, 3)$-Ramsey graphs and the Folkman number $F_e(3, 3; 4)$

TL;DR

This paper establishes an upper bound on the independence number of certain Ramsey graphs and uses this to improve the lower bound of the Folkman number F_e(3, 3; 4) from 20 to 21 through theoretical and computational methods.

Contribution

It proves a new upper bound on the independence number of (3,3)-Ramsey graphs without 4-cliques and improves the lower bound of the Folkman number F_e(3, 3; 4).

Findings

Proved that for such graphs, lpha(G)  n - 16.

Established that F_e(3, 3; 4)  21.

Used computational methods to support the new bound.

Abstract

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 $F_e(3, 3; 4)$. The number $F_e(3, 3; 4)$ is referred to as the most wanted Folkman number. It is known that $20 \leq F_e(3, 3; 4) \leq 786$. In this paper we prove that if $G$ is an $n$-vertex $(3, 3)$-Ramsey graph without 4-cliques, then $\alpha(G) \leq n - 16$, where $\alpha(G)$ denotes the independence number of $G$. Using the newly obtained bound on $\alpha(G)$ and complex computer calculations we obtain the new lower bound $$F_e(3, 3; 4) \geq 21.$$

Tables

Table 1. Table 1 : History of the Folkman number F e ​ ( 3 , 3 ; 4 ) subscript 𝐹 𝑒 3 3 4 F_{e}(3,3;4) from [ 14 ]

year	lower/upper	who/what
	bounds
1967	any?	Erdős and Hajnal [7]
1970	exist	Folkman [8]
1972	11 –	Lin implicit in [16], implied by $F_e(3,3;5)\ge 10$
1975	– $10\times {10}^{10}?$	Erdős offers $100 for proof [6]
1983	13 –	implied by a result of Nenov [23]
1984	14 –	implied by a result of Nenov [24]
1986	– $8\times {10}^{11}$	Frankl and Rödl [9]
1988	– $3\times {10}^9$	Spencer [29]
1999	16 –	Piwakowski, Radziszowski, and Urbański, implicit in [25]
2007	19 –	Radziszowski and Xu [27]
2008	– 9697	Lu [17]
2008	– 941	Dudek and Rödl [5]
2012	– 100?	Graham offers $100 for proof
2014	– 786	Lange, Radziszowski, and, Xu [15]
2017	20 –	Bikov and Nenov [4]

Table 2. Table 2 : Some properties of the graphs in ℒ ​ ( 14 ; 1 ) ℒ 14 1 \mathcal{L}(14;1) obtained in [ 25 ]

$\left|E\left(G\right)\right|$ $\#$		$\delta \left(G\right)$ $\#$		$\Delta \left(G\right)$ $\#$		$\alpha \left(G\right)$ $\#$
42	1	4	91	7	3	4	111
43	2	5	58	8	90	5	39
44	7	6	4	10	60	6	2
45	20					7	1
46	37
47	45
48	28
49	11
50	2

Table 3. Table 4 : Some properties of the graphs in ℒ ( 8 ; 3 ; ≤ 4 ) \mathcal{L}(8;3;\leq 4)

$\left|E\left(G\right)\right|$ $\#$		$\delta \left(G\right)$ $\#$		$\Delta \left(G\right)$ $\#$		$\alpha \left(G\right)$ $\#$
10	1	1	15	3	2	2	3
11	3	2	552	4	108	3	705
12	28	3	560	5	610	4	470
13	114	4	49	6	387
14	258	5	2	7	71
15	328
16	253
17	127
18	47
19	14
20	4
21	1

Table 4. Table 6 : Some properties of the graphs in ℒ + K 3 ​ ( 16 ; 1 ; 4 ) subscript ℒ subscript 𝐾 3 16 1 4 \mathcal{L}_{+K_{3}}(16;1;4)

$\left|E\left(G\right)\right|$ $\#$		$\delta \left(G\right)$ $\#$		$\Delta \left(G\right)$ $\#$
48	1	3	2782333	7	426
49	41	4	248294425	8	269602932
50	1263	5	1961917314	9	3080309372
51	24897	6	1627736506	10	535664232
52	340818	7	51394620	11	6544240
53	3215961	8	1676	12	5672
54	20943254
55	94567255
56	295234663
57	632937375
58	926347803
59	921306723
60	619034510
61	278204812
62	82280578
63	15662269
64	1876177
65	141052
66	7088
67	314
68	18
69	2

Equations

$$F_e(3,3;4)\ge 21.$$

$$20\le F_e(3,3;4)\le 786,\text{cite}[cite][\text{@@bibref}BN17]\text{cite}[cite][\text{@@bibref}LRX14].$$

$$\alpha (G)\le n-16.$$

$$\min \{\chi (G):G\in {\mathcal{H}}_e(3,3;q)\}\ge 6,\text{cite}[cite][\text{@@bibref}Lin72].$$

$$\min \{\delta (G):G\text{mbox}isavertex-criticalgraphin{\mathcal{H}}_e(3,3;4)\}\ge 8,\text{cite}[cite][\text{@@bibref}Bik18]\text{cite}[cite][\text{@@bibref}Bik16],$$

$$G\in \mathcal{L}(n;p)\Rightarrow \chi (G)\ge 6-p.$$

$$\mathcal{M}(H)=\{M_1,...,M_k\}.$$

$$N_{B(H)}(u_i)=M_i,i=1,...,k.$$

$$N_{\tilde{G}}(v_i)\in \mathcal{M}(H),i=1,...,s.$$

$$\tilde{G}\stackrel{e}{\to }(3,3).$$

$$(x_i\vee x_j\vee x_k)\wedge (\overline{x_i}\vee \overline{x_j}\vee \overline{x_k}).$$

$$\alpha (G)\le n-17.$$

Full text

In this paper we prove that if $G$ is an $n$-vertex $(3,3)$-Ramsey graph without 4-cliques, then $\alpha(G)\leq n-16$, where $\alpha(G)$ denotes the independence number of $G$. Using the newly obtained bound on $\alpha(G)$ and complex computer calculations we obtain the new lower bound

[TABLE]

*Keywords: * $(3,3)$-Ramsey graph, Folkman number, clique number, independence number

1 Introduction

Only simple graphs are considered. Let $a_{1},...,a_{s}$ be positive integers. The symbol $G\overset{e}{\rightarrow}(a_{1},...,a_{s})$ ($G\overset{v}{\rightarrow}(a_{1},...,a_{s})$) means that for every coloring of the edges (vertices) of the graph $G$ in $s$ colors there exist $i\in\{1,...,s\}$ such that there is a monochromatic $a_{i}$-clique of color $i$. If $G\overset{e}{\rightarrow}(3,3)$ we say that $G$ is a $(3,3)$-Ramsey graph.

Define:

$\mathcal{H}_{e}(a_{1},...,a_{s};q)=\{G:G\overset{e}{\rightarrow}(a_{1},...,a_{s})\mbox{ and }\omega(G)<q\}.$

$\mathcal{H}_{e}(a_{1},...,a_{s};q;n)=\{G:G\in\mathcal{H}_{e}(a_{1},...,a_{s};q)\mbox{ and }|\operatorname{V}(G)|=n\}.$

The edge Folkman numbers $F_{e}(a_{1},...,a_{s};q)$ are defined with the equality

$F_{e}(a_{1},...,a_{s};q)=\min\{|\operatorname{V}(G)|:G\in\mathcal{H}_{e}(a_{1},...,a_{s};q)\},$

i.e. $F_{e}(a_{1},...,a_{s};q)$ is the smallest positive integer $n$ for which $\mathcal{H}_{e}(a_{1},...,a_{s};q;n)\neq\emptyset$. These notations are first defined in [18], where some important properties of the Folkman numbers are proved.

Folkman [8] proved in 1970 that

$\mathcal{H}_{e}(a_{1},a_{2};q)\neq\emptyset\Leftrightarrow q\geq\max\{a_{1},a_{2}\}+1.$

Therefore, $F_{e}(3,3;q)$ exists if and only if $q\geq 4$.

From $K_{6}\overset{e}{\rightarrow}(3,3)$ it follows that $F_{e}(3,3;q)=6$ if $q\geq 7$. It is also known that

F_{e}(3,3;q)=\begin{cases}8,&\emph{if q=6, \cite[cite]{[\@@bibref{}{Gra68}{}{}]}}\\ 15,&\emph{if q=5, \cite[cite]{[\@@bibref{}{Nen81}{}{}]} and \cite[cite]{[\@@bibref{}{PRU99}{}{}]}}.\end{cases}

The exact value of the number $F_{e}(3,3;4)$ is not yet computed. For now it is known that

[TABLE]

In Table 1 are given the main stages in bounding $F_{e}(3,3;4)$

More information about the numbers $F_{e}(3,3;q)$ can be found in [11], [14], [15] and [28]. As seen on Table 1, the number $F_{e}(3,3;4)$ is very hard to bound and it is the most searched Folkman number. The reason for this is that we know very little about the graphs in $\mathcal{H}_{e}(3,3;4)$.

In this work we give an upper bound on the independence number of the graphs in $\mathcal{H}_{e}(3,3;4)$ by proving the following

Theorem 1.1**.**

Let $G\in\mathcal{H}_{e}(3,3;4;n)$. Then

[TABLE]

With the help of computer calculations and Theorem 1.1 we improve the main result $F_{e}(3,3;4)\geq 20$ from [4] by proving:

Theorem 1.2**.**

$F_{e}(3,3;4)\geq 21$.

2 Some necessary properties of the graphs in $\mathcal{H}_{e}(3,3;q)$

Many useful properties of the graphs in $\mathcal{H}_{e}(3,3;q)$ follow from the fact that homomorphism of graphs preserves the Ramsey properties. In our situation, this means

Proposition 2.1**.**

Let $G\overset{\phi}{\rightarrow}G^{\prime}$ be a homomorphism and $G\overset{e}{\rightarrow}(3,3)$. Then, $G^{\prime}\overset{e}{\rightarrow}(3,3)$.

Proof.

Suppose the opposite is true and consider a 2-coloring of $\operatorname{E}(G^{\prime})$ without monochromatic triangles. Define a 2-coloring of $\operatorname{E}(G)$ in the following way: the edge $[u,v]$ is colored in the same color as the edge $[\phi(u),\phi(v)]$. Clearly, this coloring of $\operatorname{E}(G)$ does not contain monochromatic triangles. ∎

In the general case, it is true that $G\overset{e}{\rightarrow}(a_{1},...,a_{s})\Rightarrow G^{\prime}\overset{e}{\rightarrow}(a_{1},...,a_{s})$, and $G\overset{v}{\rightarrow}(a_{1},...,a_{s})\Rightarrow G^{\prime}\overset{v}{\rightarrow}(a_{1},...,a_{s})$, which is proved in the same way.

Now consider the canonical homomorphism $G\overset{\phi}{\rightarrow}K_{\chi(G)}$. If $G\overset{e}{\rightarrow}(3,3)$, then $K_{\chi(G)}\overset{e}{\rightarrow}(3,3)$, and therefore

[TABLE]

For $q\geq 5$, the inequality (2.1) is exact. It is not known whether this inequality is exact in the case $q=4$. This is a special case of a result of Lin [16]. In the general case, $G\overset{e}{\rightarrow}(a_{1},...,a_{s})\Rightarrow\chi(G)\geq R(a_{1},...,a_{s})$, which is proved in the same way.

With $K_{p}+G$ we denote the graph obtained by connecting with an edge every vertex of $K_{p}$ to every vertex of $G$.

Further, we will need the following

Proposition 2.2**.**

Let $G\overset{e}{\rightarrow}(3,3)$, $A$ be an independent set of vertices of $G$, and $H=G-A$. Then, $K_{1}+H\overset{e}{\rightarrow}(3,3)$.

Proof.

Consider the mapping $G\overset{\phi}{\rightarrow}K_{1}+H$:

\phi(v)=\begin{cases}\operatorname{V}(K_{1}),&\emph{if v\in A}\\ v,&\emph{if v\in\operatorname{V}(H)}.\end{cases}

It is clear that $\phi$ is a homomorphism, and according to Proposition 2.1, $K_{1}+H\overset{e}{\rightarrow}(3,3)$. ∎

The usefulness of Proposition 2.2 lies in the fact that the graph $G$ can be obtained by adding independent vertices to the smaller graph $H$. In the general case it is true that if $G\overset{e}{\rightarrow}(a_{1},...,a_{s})$, then $K_{1}+H\overset{e}{\rightarrow}(a_{1},...,a_{s})$.

Remark 2.3**.**

Other proof of Proposition 2.2 is given in the proof of Theorem 1.2 from [27]. However, the proposition is not explicitly formulated.

A topic of significant interest are homomorphisms in Proposition 2.1 which do not increase the clique number. They could be used to obtain non trivial results. For example, in [21] a 20-vertex graph in $\mathcal{H}_{e}(3,3;5)$ is constructed. Using a homomorphism, in the same work a 16-vertex graph in $\mathcal{H}_{e}(3,3;5)$ is obtained from this graph. Thus, in 1978 the bound $F_{e}(3,3;5)\leq 16$ is proved, improving the previous result $F_{e}(3,3;5)\leq 18$ from 1973 [13].

The graph $G$ is vertex-critical (edge-critical) in $\mathcal{H}_{e}(3,3;4)$ if $G\in\mathcal{H}_{e}(3,3;4)$ and $G-v\not\in\mathcal{H}_{e}(3,3;4),\forall v\in\operatorname{V}(G)$ ($G-e\not\in\mathcal{H}_{e}(3,3;4),\forall e\in\operatorname{E}(G)$). It is true that

[TABLE]

where $\delta(G)$ is the minimum degree of $G$.

Remark 2.4**.**

In [2] and [3] (2.2) is formulated for edge-critical graphs without isolated vertices. The proof is obviously also true for vertex critical graphs, and therefore further we shall use (2.2).

It is not known if the inequality (2.2) is exact.

3 Auxiliary notation and propositions

Let $G\in\mathcal{H}_{e}(3,3;4)$, $A$ be an independent set of vertices of $G$, and $H_{1}=G-A$. By Proposition 2.2, $K_{1}+H_{1}\overset{e}{\rightarrow}(3,3)$. If $A_{1}$ is an independent set in $H_{1}$ and $H_{2}=H_{1}-A_{1}$, then $K_{2}+H_{2}\overset{e}{\rightarrow}(3,3)$. If $A_{2}$ is an independent set in $H_{2}$ and $H_{3}=H_{2}-A_{2}$, then $K_{3}+H_{3}\overset{e}{\rightarrow}(3,3)$, etc. This way, we obtain a sequence $G\supseteq H_{1}\supseteq H_{2}\supseteq H_{3}\supseteq...$, in which $\omega(H_{i})\leq 3$ and $K_{i}+H_{i}\overset{e}{\rightarrow}(3,3)$. Further, in the proof of Theorem 1.2, we will use such a sequence of graphs. Because of this, the following notations are convenient:

$\mathcal{L}(n;p)=\left\{G:|\operatorname{V}(G)|=n,\omega(G)<4\mbox{ and }K_{p}+G\overset{e}{\rightarrow}(3,3)\right\}$

$\mathcal{L}(n;p;s)=\left\{G\in\mathcal{L}(n;p):\alpha(G)=s\right\}$

Obviously, $\mathcal{L}(n;0)=\mathcal{H}_{e}(3,3;4;n)$. Let us note that

$\mathcal{L}(n;1)=\emptyset$, if $n\leq 13$, [25].

$|\mathcal{L}(14;1)|=153$, [25].

$|\mathcal{L}(15;1)|=2081234$, [4].

The graphs in $\mathcal{L}(16;1)$ are not known. In the proof of Theorem 1.2 we obtain some of the graphs in $\mathcal{L}(16;1)$. The graphs in $\mathcal{L}(15;1)$ will be used in the proofs of Theorem 1.1 and Theorem 1.2. Some properties of the graphs in $\mathcal{L}(14;1)$, $\mathcal{L}(15;1)$, and $\mathcal{L}(16;1)$ are given in Table 3, Table 3, and Table 7.

It is true that

$G\overset{v}{\rightarrow}(3,3)\Rightarrow K_{1}+G\overset{e}{\rightarrow}(3,3)$, ([13], see acknowledgments).

This fact is known as Posa’s implication. Posa uses this implication to prove that $\mathcal{H}_{e}(3,3;5)\neq\emptyset$ (unpublished). Irwing [13] uses the implication to prove the bound $\mathcal{H}_{e}(3,3;5)\leq 18$. If additionally $\omega(G)=3$, then $G\in\mathcal{L}(n,1)$. In [25] it is proved that if $G\in\mathcal{L}(14,1)$, then $G\overset{v}{\rightarrow}(3,3)$. This result was used in [27] to obtain the bound $F_{e}(3,3;4)\geq 19$. There exist, however, graphs $G$ in $\mathcal{L}(15,1)$ which do not have the property $G\overset{v}{\rightarrow}(3,3)$. There are 20 such graphs and they are obtained in [4]. Furthermore, these graphs do not have the property $G\overset{v}{\rightarrow}(2,2,3)$. This is one of the reasons why the method in the proof of $F_{e}(3,3;4)\geq 19$ in [27] is inapplicable for proving $F_{e}(3,3;4)\geq n$, $n\geq 20$.

By Proposition 2.2, if $G\in\mathcal{L}(n,0)$ and $A$ is an independent set of vertices of $G$, then $G-A\in\mathcal{L}(n-|A|,1)$. In [4] we formulate without proof the following generalization of this fact:

Proposition 3.1**.**

[4]** Let $G\in\mathcal{L}(n;p)$, $A\subseteq\operatorname{V}(G)$ be an independent set of vertices of $G$ and $H=G-A$. Then $H\in\mathcal{L}(n-|A|;p+1)$.

Proof.

Since, $G\in\mathcal{L}(n;p)$, $K_{p}+G\overset{e}{\rightarrow}(3,3)$. According to Proposition 2.2, $K_{1}+((K_{p}+G)-A)\overset{e}{\rightarrow}(3,3)$. Since $(K_{p}+G)-A=K_{p}+(G-A)=K_{p}+H$ and $K_{1}+(K_{p}+H)=K_{p+1}+H$, we obtain $K_{p+1}+H\overset{e}{\rightarrow}(3,3)$. Thus, $H\in\mathcal{L}(n-|A|;p+1)$. ∎

We denote by $\mathcal{L}_{max}(n;p;s)$ the set of all maximal $K_{4}$-free graphs in $\mathcal{L}(n;p;s)$, i.e. the graphs $G\in\mathcal{L}(n;p;s)$ for which $\omega(G+e)=4$ for every $e\in\operatorname{E}(\overline{G})$.

The graph $G$ is called a $(+K_{3})$-graph if $G+e$ contains a new $3$-clique for every $e\in\operatorname{E}(\overline{G})$. Clearly, $G$ is a $(+K_{3})$-graph if and only if $N(u)\cap N(v)=\emptyset$ for every pair of non-adjacent vertices $u$ and $v$ of $G$, i.e. either $G$ is a complete graph or the diameter of $G$ is equal to 2. The set of all $(+K_{3})$-graphs in $\mathcal{L}(n;p;s)$ is denoted by $\mathcal{L}_{+K_{3}}(n;p;s)$. Obviously, $\mathcal{L}_{max}(n;p;s)\subseteq\mathcal{L}_{+K_{3}}(n;p;s)$.

For convenience, we will also use the following notations:

$\mathcal{L}_{max}(n;p;\leq s)=\bigcup_{s^{\prime}\leq s}\mathcal{L}_{max}(n;p;s^{\prime})$

$\mathcal{L}_{+K_{3}}(n;p;\leq s)=\bigcup_{s^{\prime}\leq s}\mathcal{L}_{+K_{3}}(n;p;s^{\prime})$

It is easy to see that if $G$ is a maximal $K_{4}$-free graph and $A$ is an independent set of vertices in $G$, then $G-A$ is a $(+K_{3})$-graph. Because of this, regarding the graphs in $\mathcal{L}_{max}(n;p;s)$, from Proposition 3.1 it follows easily that

Proposition 3.2**.**

[4]** Let $G\in\mathcal{L}_{max}(n;p;s)$. Let $A\subseteq\operatorname{V}(G)$ be an independent set of vertices of $G$, $|A|=s$ and $H=G-A$. Then,

$H\in\mathcal{L}_{+K_{3}}(n-s;p+1;\leq s)$.

Further, the bound $F_{e}(3,3;4)\geq 21$ will be proved with the help of Algorithm 5.1 and Algorithm 5.3, which are based on Proposition 3.2.

Definition 3.3**.**

The graph $G$ is called a Sperner graph if $N_{G}(u)\subseteq N_{G}(v)$ for some pair of vertices $u,v\in\operatorname{V}(G)$.

Let $G\in\mathcal{L}(n;p;s)$ and $N_{G}(u)\subseteq N_{G}(v)$. Then, $K_{p}+(G-u)$ is a homomorphic image of $K_{p}+G$ and by Proposition 2.1, $K_{p}+(G-u)\overset{e}{\rightarrow}(3,3)$, i.e. $G-u\in\mathcal{L}(n-1;p;s^{\prime})$, $s-1\leq s^{\prime}\leq s$. Therefore, every Sperner graph $G\in\mathcal{L}(n;p;s)$ is obtained by adding one new vertex to some graph $H\in\mathcal{L}(n-1;p;s^{\prime})$, $s-1\leq s^{\prime}\leq s$. In the special case, when $G$ is a Sperner graph and $G\in\mathcal{L}_{max}(n;p;s)$, from $N_{G}(u)\subseteq N_{G}(v)$ it follows that $N_{G}(u)=N_{G}(v)$ and $G-u\in\mathcal{L}_{max}(n-1;p;s^{\prime})$, $s-1\leq s^{\prime}\leq s$. Hence, it is true

Proposition 3.4**.**

If $G\in\mathcal{L}_{max}(n;p;s)$ is a Sperner graph, then $G$ is obtained by duplicating a vertex in some graph $H\in\mathcal{L}_{max}(n-1;p;s^{\prime})$, $s-1\leq s^{\prime}\leq s$.

From (2.1) and $K_{p}+G\overset{e}{\rightarrow}(3,3)$ it follows that

[TABLE]

We will use this fact in Algorithm 5.1.

4 Proof of Theorem 1.1

Definition 4.1**.**

For every graph $H$ denote by $\mathcal{M}(H)$ the set of all maximal $K_{3}$-free subsets of $\operatorname{V}(H)$. Let

[TABLE]

We denote by $\operatorname{B}(H)$ the graph which is obtained by adding to $H$ new independent vertices $u_{1},...,u_{k}$ and new edges incident to $u_{1},...,u_{k}$ such that

[TABLE]

Lemma 4.2**.**

Let $G$ be a graph, $\omega(G)=3$, $A$ be an independent set of vertices of $G$, and $H=G-A$. If $G\overset{e}{\rightarrow}(3,3)$, then $\operatorname{B}(H)\overset{e}{\rightarrow}(3,3)$.

Proof.

Let $\mathcal{M}(H)=\{M_{1},...,M_{k}\}$ be the same as in Definition 4.1 and $A=\{v_{1},...,v_{s}\}$. Let $v_{i}\in A$. Then, $N_{G}(v_{i})\subseteq M_{j}$ for some $j\in\{1,...,k\}$. Let $j_{i}$ be the smallest index for which $N_{G}(v_{i})\subseteq M_{j_{i}}$. We define a supergraph $\widetilde{G}$ of $G$ in the following way: for each $v_{i}\in A$ we add to $\operatorname{E}(G)$ the new edges $[v_{i},u],u\in M_{j_{i}}\setminus N_{G}(v_{i})$. Clearly, $\operatorname{V}(\widetilde{G})=\operatorname{V}(G)$, $A$ is an independent set of vertices of $\widetilde{G}$, $\widetilde{G}-A=H$ and

[TABLE]

Since $\widetilde{G}$ is a subgraph of $G$, it follows

[TABLE]

If $\{N_{\widetilde{G}}(v_{1}),...,N_{\widetilde{G}}(v_{s})\}$ is a subset of $\mathcal{M}(H)$, then $\widetilde{G}$ is a subgraph of $\operatorname{B}(H)$ and from (4.1) it follows that $\operatorname{B}(H)\overset{e}{\rightarrow}(3,3)$.

Let $\{N_{\widetilde{G}}(v_{1}),...,N_{\widetilde{G}}(v_{s})\}$ be a multiset and $N_{\widetilde{G}}(v_{i})=N_{\widetilde{G}}(v_{j}),\ i<j$. Let $\widetilde{G^{\prime}}$ = $\widetilde{G}-v_{j}$. Obviously, $\widetilde{G^{\prime}}$ is a homomorphic image of $G$. Therefore, from Proposition 2.1 and (4.1) it follows that $\widetilde{G^{\prime}}\overset{e}{\rightarrow}(3,3)$.

If in $\{N_{\widetilde{G^{\prime}}}(v_{i})|i=1,...s,i\neq j\}$ there is also a duplication, then in the same way we remove from $\widetilde{G^{\prime}}$ one of the duplicating vertices among $v_{1},...,v_{i-1},v_{i+1},...,v_{s}$ and we obtain a graph $\widetilde{G^{\prime\prime}}$ such that $\widetilde{G^{\prime\prime}}\overset{e}{\rightarrow}(3,3)$.

In the end, a graph $\widetilde{\widetilde{G}}$ is reached such that $H=\widetilde{\widetilde{G}}-A^{\prime}$, where $A^{\prime}\subseteq A$, $\{N_{\widetilde{\widetilde{G}}}(v)|v\in A^{\prime}\}$ is a subset of $\mathcal{M}(H)$, and $\widetilde{\widetilde{G}}\overset{e}{\rightarrow}(3,3)$. Since $\widetilde{\widetilde{G}}$ is a subgraph of $\operatorname{B}(H)$, it follows that $\operatorname{B}(H)\overset{e}{\rightarrow}(3,3)$. ∎

*Proof of *Theorem 1.1. Suppose the opposite is true, i.e. $\alpha(G)\geq n-15$. Let $A=\{v_{1},...,v_{n-15}\}$ be an independent set of vertices of $G$, and $H=G-A$. Then, from Lemma 4.2 it follows $\operatorname{B}(H)\overset{e}{\rightarrow}(3,3)$.

According to Proposition 3.1 ($p=0$), $H\in\mathcal{L}(15;1)$. All 2081234 graphs in $\mathcal{L}(15;1)$ were obtained in ([4], Remark 4.4). With a computer we check that $\operatorname{B}(H)\not\overset{e}{\rightarrow}(3,3)$ for all $H\in\mathcal{L}(15,1)$. This contradiction proves the theorem. ∎

Corollary 4.3**.**

[4]** $F_{e}(3,3;4)\geq 20$.

Proof.

Suppose that $G$ is a 19-vertex $(3,3)$-Ramsey graph and $\omega(G)=3$. From Theorem 1.1 it follows that $\alpha(G)\leq 3$. This contradicts with the equality $R(4,4)=18$. ∎

The graphs $\operatorname{B}(H),H\in\mathcal{L}(15;1)$, in the proof of Theorem 1.1 have between 50 and 210 vertices. We used the zchaff SAT solver [30] to prove that these graphs are not $(3,3)$-Ramsey graphs. The problem of determining if a graph $G$ is a $(3,3)$-Ramsey graph can be transformed into a problem of satisfiability of a boolean formula $\phi_{G}$ in conjunctive normal form with $|\operatorname{E}(G)|$ variables. Let $e_{1},...,e_{|\operatorname{E}(G)|}$ be the edges of $G$ and $x_{1},...,x_{|\operatorname{E}(G)|}$ be the corresponding boolean variables in $\phi_{G}$. For every triangle in $G$ formed by the edges e_{i}$$e_{j}$$e_{k}, we add two clauses to $\phi_{G}$

[TABLE]

It is easy to see that $G\overset{e}{\rightarrow}(3,3)$ if and only if $\phi_{G}$ is satisfiable.

Even though the graphs $\operatorname{B}(H),H\in\mathcal{L}(15;1)$, have up to 210 vertices, SAT solvers are able to solve the resulting boolean formulas in a short amount of time. There exist smaller graphs $G$ for which it is difficult to determine if $G\overset{e}{\rightarrow}(3,3)$. For example, Exoo conjectured that the 127-vertex graph $G_{127}$, used by Hill and Irwing [12] to prove the bound $R(4,4,4)\geq 128$, has the property $G_{127}\overset{e}{\rightarrow}(3,3)$. This conjecture was studied in [27] and [14]. It is still unknown whether $G_{127}\overset{e}{\rightarrow}(3,3)$.

5 Proof of Theorem 1.2

According to Proposition 3.4, all Sperner graphs in $\mathcal{L}_{max}(n;p;s)$ can be obtained easily by duplicating a vertex in graphs in $\mathcal{L}_{max}(n-1;p;s^{\prime}),s-1\leq s^{\prime}\leq s$. By Proposition 3.2, the non-Sperner graphs in $\mathcal{L}_{max}(n;p;s)$ are obtained by adding $s$ independent vertices to some graphs in $\mathcal{L}_{+K_{3}}(n-s;p+1;\leq s)$. This is realized with the help of the following algorithm:

Algorithm 5.1**.**

[4]** Finding all non-Sperner graphs in $\mathcal{L}_{max}(n;p;s)$ for fixed $n$, $p$ and $s$.

1.* The input of the algorithm is the set $\mathcal{A}=\mathcal{L}_{+K_{3}}(n-s;p+1;\leq s)$. The output will be the set $\mathcal{B}$ of all non-Sperner graphs in $\mathcal{L}_{max}(n;p;s)$. Initially, set $\mathcal{B}=\emptyset$.*

2.* For each graph $H\in\mathcal{A}$:*

2.1.* Find the family $\mathcal{M}(H)=\left\{M_{1},...,M_{t}\right\}$ of all maximal $K_{3}$-free subsets of $\operatorname{V}(H)$.*

2.2.* Find all $s$-element subsets $N=\left\{M_{i_{1}},M_{i_{2}},...,M_{i_{s}}\right\}$ of $\mathcal{M}(H)$, which fulfill the conditions:*

(a) $M_{i_{j}}\neq N_{H}(v)$ for every $v\in\operatorname{V}(H)$ and for every $M_{i_{j}}\in N$.

(b) $K_{2}\subseteq M_{i_{j}}\cap M_{i_{k}}$ for every $M_{i_{j}},M_{i_{k}}\in N$.

(c) $\alpha(H-\bigcup_{M_{i_{j}}\in N^{\prime}}M_{i_{j}})\leq s-|N^{\prime}|$ for every $N^{\prime}\subseteq N$.

2.3.* For each of the found in 2.2 $s$-element subsets $N=\left\{M_{i_{1}},M_{i_{2}},...,M_{i_{s}}\right\}$ of $\mathcal{M}(H)$ construct the graph $G=G(N)$ by adding new independent vertices $v_{1},v_{2},...,v_{s}$ to $\operatorname{V}(H)$ such that $N_{G}(v_{j})=M_{i_{j}},j=1,...,s$. If $G$ is not a Sperner graph and $\omega(G+e)=4,\forall e\in\operatorname{E}(\overline{G})$, then add $G$ to $\mathcal{B}$.*

3.* Remove the isomorph copies of graphs from $\mathcal{B}$.*

4.* Remove from $\mathcal{B}$ all graphs with chromatic number less than $6-p$.*

5.* Remove from $\mathcal{B}$ all graphs $G$ for which $K_{p}+G\not\overset{e}{\rightarrow}(3,3)$.*

Theorem 5.2**.**

[4]** After the execution of Algorithm 5.1, the obtained set $\mathcal{B}$ coincides with the set of all non-Sperner graphs in $\mathcal{L}_{max}(n;p;s)$.

The correctness of Algorithm 5.1 is guarantied by the proof of Theorem 5.2 in [4]. Here we will only note some details. The condition (a) has to be satisfied since $G=G(N)$ is not a Sperner graph. The condition (b) follows from the maximality of $G=G(N)$. Even if both conditions (a) and (b) are satisfied, additional checks in step 2.3 are still needed to make sure that only maximal non-Sperner graphs are added to $\mathcal{B}$. From condition (c) it follows that only graphs with independence number $s$ are added to $\mathcal{B}$. If $N^{\prime}=\emptyset$, then (c) clearly holds, since $\alpha(H)\leq s$. If $|N^{\prime}|=1$, then for each added vertex $v_{j}$ it is guarantied that $v_{j}$ does not form an independent set with $s$ vertices of $H$. If $|N^{\prime}|=2$, then for every two added vertices $v_{j},v_{k}$ it is guarantied that $v_{j}$ and $v_{k}$ do not form an independent set with $(s-1)$ vertices of $H$, etc. The graphs in $\mathcal{B}$ must satisfy the condition in step 4 according to (3.1).

According to Corollary 4.3, the graphs in $\mathcal{L}(20;0)$ are vertex critical, and by (2.2), these graphs have minimal degree greater than or equal to 8. Because of this, we modify Algorithm 5.1 in the case $p=0$ in such a way that only graphs $G$ with $\delta(G)\geq 8$ are added to $\mathcal{B}$:

Algorithm 5.3**.**

[4]** Optimization of Algorithm 5.1 for finding all non-Sperner graphs $G\in\mathcal{L}_{max}(n;0;s)$ with $\delta(G)\geq 8$.

1. In step 1 we remove from the set $\mathcal{A}$ the graphs with minimum degree less than 8 - s.

2. In step 2.2 we add the following conditions for the subset $N$:

(d) $|M_{i_{j}}|\geq 8$ for every $M_{i_{j}}\in N$.

(e) If $N^{\prime}\subseteq N$, then $d_{H}(v)\geq 8-s+|N^{\prime}|$ for every $v\not\in\bigcup_{M_{i_{j}}\in N^{\prime}}M_{i_{j}}$.

*Proof of *Theorem 1.2. Suppose the opposite is true and let $G$ be a $20$-vertex maximal $(3,3)$-Ramsey graph with $\omega(G)=3$. From Theorem 1.1 it follows that $\alpha(G)\leq 4$. Now, from $R(4,4)=18$ it follows that $\alpha(G)=4$. Therefore, it is enough to prove that $\mathcal{L}_{max}(20;0;4)=\emptyset$. First, we will successively obtain all graphs in the sets $\mathcal{L}_{+K_{3}}(8;3;\leq 4)$, $\mathcal{L}_{+K_{3}}(12;2;\leq 4)$, and $\mathcal{L}_{+K_{3}}(16;1;\leq 4)$.

Obtaining all graphs in $\mathcal{L}_{+K_{3}}(8;3;\leq 4)$:

We use the geng tool included in the nauty programs [20] to generate All non-isomorphic graphs of order 8 are generated using the geng tool included in the nauty programs [20]. Among them we find all 1178 graphs in $\mathcal{L}_{+K_{3}}(8;3;\leq 4)$ (see Table 5).

Obtaining all graphs in $\mathcal{L}_{+K_{3}}(12;2;\leq 4)$:

From $R(3,4)=9$ it follows that $\mathcal{L}(12;2;\leq 2)=\emptyset$. All 1449166 12-vertex graphs $G$ with $\omega(G)<4$ and $\alpha(G)<4$ are known and available on [19]. Among them there are 321 graphs in $\mathcal{L}_{max}(12;2;3)$. We use geng to generate all non-isomorphic graphs of order 11. Among them we find all 372 graphs in $\mathcal{L}_{max}(11;2;\leq 4)$. According to Proposition 3.4, all Sperner graphs in $\mathcal{L}_{max}(12;2;4)$ are obtained by duplicating a vertex in some of the graphs in $\mathcal{L}_{max}(11;2;\leq 4)$. This way, we find all 1341 Sperner graphs in $\mathcal{L}_{max}(12;2;4)$. We execute Algorithm 5.1 ($n=12,p=2,s=4$) with input the set $\mathcal{A}=\mathcal{L}_{+K_{3}}(8;3;\leq 4)$ to output all 815 non-Sperner graphs in $\mathcal{L}_{max}(12;2;4)$. Thus, $|\mathcal{L}_{max}(12;2;\leq 4)|=2477$. By removing edges from the graphs in $\mathcal{L}_{max}(12;2;\leq 4)$ we find all 539410034 graphs in $\mathcal{L}_{+K_{3}}(12;2;\leq 4)$ (see Table 5).

Obtaining all graphs in $\mathcal{L}_{+K_{3}}(16;1;\leq 4)$:

From $R(3,4)=9$ it follows that $\mathcal{L}(16;1;\leq 2)=\emptyset$. There are only two 16-vertex graphs $G$ such that $\omega(G)<4$ and $\alpha(G)<4$ [19]. We check with a computer that none of them belongs to $\mathcal{L}(16;1)$, and therefore $\mathcal{L}(16;1;3)=\emptyset$. Thus, $\mathcal{L}(16;1;\leq 4)=\mathcal{L}(16;1;4)$. All 5772 graphs in $\mathcal{L}_{max}(15;1;\leq 4)$ were obtained in part 1 of the proof of the Main Theorem in [4]. According to Proposition 3.4, all Sperner graphs in $\mathcal{L}_{max}(16;1;4)$ are obtained by duplicating a vertex in some of the graphs in $\mathcal{L}_{max}(15;1;\leq 4)$. This way, we find all 21749 Sperner graphs in $\mathcal{L}_{max}(16;1;4)$. We execute Algorithm 5.1 ($n=16,p=1,s=4$) with input the set $\mathcal{A}=\mathcal{L}_{+K_{3}}(12;2;\leq 4)$ to output all 1676267 non-Sperner graphs in $\mathcal{L}_{max}(16;1;4)$. Thus, $|\mathcal{L}_{max}(16;1;\leq 4)|=1698016$. By removing edges from the graphs in $\mathcal{L}_{max}(16;1;\leq 4)$ we find all 3892126874 graphs in $\mathcal{L}_{+K_{3}}(16;1;\leq 4)$ (see Table 7).

Proving that $\mathcal{L}_{max}(20;0;4)=\emptyset$:

We execute Algorithm 5.3 ($n=20,p=0,s=4$) with input the set $\mathcal{A}=\mathcal{L}_{+K_{3}}(16;1;\leq 4)$. After the completion of step 4, 19803568 graphs remain in the set $\mathcal{B}$ (see Table 7). None of these graphs satisfies the condition in step 5, hence $\mathcal{B}=\emptyset$. We obtained that there are no non-Sperner graphs in $\mathcal{L}_{max}(20;0;4)$ with minimum degree greater than or equal to 8. According to Corollary 4.3, all graphs in $\mathcal{L}_{max}(20;0;4)$ must be vertex critical. Therefore, there are no Sperner graphs in $\mathcal{L}_{max}(20;0;4)$, and by (2.2), no graphs with minimum degree less than 8. We proved that $\mathcal{L}_{max}(20;0;4)=\emptyset$, which finishes the proof. ∎

Some properties of the graphs in $\mathcal{L}_{+K_{3}}(8;3;\leq 4)$, $\mathcal{L}_{+K_{3}}(12;2;\leq 4)$, and $\mathcal{L}_{+K_{3}}(16;1;4)$ are given in Table 5, Table 5, and Table 7. Properties of the 20-vertex graphs obtained after the completion of step 4 of Algorithm 5.3 $(n=20,p=0,s=4)$ are given in Table 7.

All computations were performed on a personal computer. The most time consuming part of the proof was obtaining all graphs in $\mathcal{L}_{+K_{3}}(16;1;4)$ by removing edges from the graphs in $\mathcal{L}_{max}(16;1;4)$, which took about 4 months. After that, executing Algorithm 5.3 $(n=20,p=0,s=4)$ with input the graphs in $\mathcal{L}_{+K_{3}}(16;1;4)$ was done in under 2 months.

In order to check the correctness of our computer programs implementing Algorithm 5.1, we reproduced the 153 graphs in $\mathcal{L}(14;1)$, which were first obtained in [25] in a different way. Among the graphs in $\mathcal{L}(14;1)$ there are 8 maximal graphs, all of which have independence number 4, i.e. $|\mathcal{L}_{max}(14;1)|=8$ and $\mathcal{L}_{max}(14;1)=\mathcal{L}_{max}(14;1;4)$. Using nauty we obtained all 547524 graphs in $\mathcal{L}_{+K_{3}}(10;2;\leq 4)$. By executing Algorithm 5.1 ($n=14,p=1,s=4$) with input $\mathcal{A}=\mathcal{L}_{+K_{3}}(10;2;\leq 4)$ we found all 8 graphs in $\mathcal{L}_{max}(14;1;4)$. By removing edges from the graphs in $\mathcal{L}_{max}(14;1)$ we obtained the 153 graphs in $\mathcal{L}(14;1)$.

6 Concluding remarks

In this section we consider the possibilities for improving the inequality $F_{e}(3,3;4)\geq 21$. We suppose the following conjecture is true:

Conjecture 6.1**.**

$\min\{\alpha(G):G\in\mathcal{H}_{e}(3,3;4)\}\geq 5$.

If $G\in\mathcal{H}_{e}(3,3;4;n)$, $n\geq 25$, according to the equality $R(4,5)=25$ we have $\alpha(G)\geq 5$. All 24-vertex graphs with independence number 4 and clique number 3 are obtained in [1]. With the help of a computer we check that none of these graphs belongs to $\mathcal{H}_{e}(3,3;4)$. This way, we proved that if $G\in\mathcal{H}_{e}(3,3;4;n)$, $n\geq 24$, then $\alpha(G)\geq 5$. To prove the conjecture it remains to consider the cases $n=21$, $22$, and $23$.

By similar reasoning as in the proof of Theorem 1.1, but with more calculations, it could be proved that

[TABLE]

From this inequality and Conjecture 6.1 it follows that $F_{e}(3,3;4)\geq 22$.

ACKNOWLEDGMENTS

We would like to thank Professor Stanisław Radziszowski for his useful remarks and suggestions which led to the improvement of this paper.

The first author was supported by the National program ”Young scientists and Postdoctoral candidates” of the Ministry of Education and Science.