1 INTRODUCTION

This work paper deals with problems in Ramsey theory. By way of introduction, we recall the definition of classical Ramsey numbers in one of the most natural cases. Let \({{K}_{n}}\) be the complete graph on n vertices (the graph is simple, i.e., it has neither loops nor multiple edges, and has no orientation either). For a simple graph G on the same \(n\) vertices, let \({{K}_{n}}{{\backslash }}G\) denote the complement of G, i.e., a spanning subgraph of Kn in which there is no edge of G, but there are all edges absent from G. An independent set of vertices in a graph is any set of its vertices, no two of which are adjacent. The cardinality of the maximum independent set of vertices in G is called the independence number of G and is denoted by \(\alpha (G)\). A clique in G is any of its complete subgraphs. The cardinality of a maximum clique in \(G\) is called the clique number of G and is denoted by \(\omega (G)\). The duality of the independence number and the clique number, which is reflected even in the notation, is easily described by the obvious equality \(\alpha (G) = \omega ({{K}_{n}}{{\backslash }}G)\). In these terms, the (diagonal) Ramsey number \(R(k)\) for a given positive integer k is defined as the minimum positive integer n such that, for any spanning subgraph G of \({{K}_{n}}\), there is a clique of size k in either G or \({{K}_{n}}{{\backslash }}G.\)

The search for classical Ramsey numbers is a major (and, at the same time, very complicated) problem on the border of graph theory and algorithm theory. Currently, it is known that (see [1, 2])

$$(1 + o(1))\frac{{\sqrt 2 }}{e}k{{2}^{{k/2}}} \leqslant R(k) \leqslant {{4}^{k}}{{e}^{{ - \gamma \frac{{{{{\ln }}^{2}}k}}{{\ln \ln k}}}}},\quad \gamma > 0.$$

Additionally, there are numerous results concerning small k (see [3]), but, in what follows, we are more interested in asymptotic assertions.

For classical Ramsey numbers, there is a huge number of important generalizations (see, e.g., [1–6]). In this paper, we focus on the formulation proposed for the first time in [7]. Let \(\{ {{G}_{n}}\} \), \(n \in \mathbb{N}\), be an arbitrary sequence of graphs. Note that \(n\) is the graph index, i.e., this is not necessarily the number of its vertices. As before, suppose that \(k \in \mathbb{N}\). Let \({{R}_{{\min }}}(\{ {{G}_{n}}\} ,k)\) denote the minimum positive integer m such that, for any spanning subgraph \(G\) of Gm, either G or \({{G}_{m}}{{\backslash }}G\) contains an induced subgraph of \({{G}_{m}}\) on k vertices. By \({{G}_{m}}{{\backslash }}G\) we mean the complement of G with respect to Gm, i.e., \({{G}_{m}}{{\backslash }}G\) has no edges of G, whereas it contains all edges of Gm that are absent in G. Edges that are not contained in \({{G}_{m}}\) cannot appear in principle.

Obviously, since a clique is an induced subgraph of Kn and there are no other induced subgraphs in the complete graph, we have \(R(k) = R(\{ {{K}_{n}}\} ,k)\). On the other hand, clearly, for many sequences of graphs, this definition makes no sense or is trivial. Moreover, for the classical case, the Ramsey property is, in a sense, monotone: if it holds for some \(n\), then it also holds for all m ≥ n; if it does not hold for some \(n\) (i.e., there is a graph on \(n\) vertices such that neither this graph nor its complement in Kn contains a k-clique), then it does not hold for all m ≤ n. Obviously, this is associated with the fact that Kn is a subgraph of each \({{K}_{m}}\) with m ≥ n. However, in the general case, this is not the case. Accordingly, it is worthwhile to consider the quantity \({{R}_{{\max }}}(\{ {{G}_{n}}\} ,k)\) defined as the maximum number \(m \in \mathbb{N}\) for which there exists a spanning subgraph \(G\) of \({{G}_{m}}\) such that neither this subgraph nor \({{G}_{m}}{{\backslash }}G\) contains induced subgraphs of \({{G}_{m}}\) on k vertices. Of course, \({{R}_{{\max }}}(\{ {{K}_{n}}\} ,k) = {{R}_{{\min }}}(\{ {{K}_{n}}\} ,k) - 1\), but, in the general case, these two quantities can be related in a different manner.

In [7] \(\{ {{G}_{n}}\} \) was specified as a sequence of graphs \(G(n,n{\text{/}}2,n{\text{/}}4)\), where \(n = 4l\) for \(l \in \mathbb{N}\); their vertices are all possible n/2-element subsets of the set \(\{ 1, \ldots ,n\} \), and vertices are adjacent if and only if the cardinality of the intersection of the corresponding sets is equal to \(n{\text{/}}4\). These graphs arise in coding theory (see [8, 9]), where they are known as Johnson graphs or schemes. They are also of crucial importance in combinatorial geometry, where they are often called distance graphs (see [9–14]). Note, finally, that a maximum clique in the graph \(G(n,n{\text{/}}2,n{\text{/}}4)\) is in direct correspondence with a classical Hadamard matrix (see [8]).

Actually, \(G(n,n{\text{/}}2,n{\text{/}}4)\) is a very special case of the more general Johnson graph \(G(n,r,s)\); here, once again, the vertices are \(r\)-element subsets of the set \(\{ 1, \ldots ,n\} \), and vertices are adjacent if and only if the cardinalities of the intersections of the corresponding sets are equal to s. These graphs play the same crucial role in coding theory and combinatorial geometry. Moreover, for s = 0, there appear so-called Kneser graphs (see [15–17]), which are a central object of modern combinatorics. Finally, it should be noted that \(G(n,1,0) = {{K}_{n}}\), i.e., the classical complete graph is a specific case in the total variety of situations.

The graphs \(G(n,r,s)\) are closely related to the graphs \(G(n,r, < s)\). The difference is that, in the latter case, vertices are adjacent if and only if the intersection of the corresponding sets has a cardinality strictly smaller than s.

In the next section, we formulate the main results and comment on them. Finally, the ideas underlying the proofs are briefly outlined in Section 3.

2 FORMULATIONS OF THE RESULTS

The following theorems hold.

Theorem 1. Suppose that \(r = r(n)\) and \(s = s(n)\) are functions such that the following conditions are satisfied:

(1) If \(r - s \geqslant 2\) for all sufficiently large \(n\), then \({{r}^{3}}\, = \,o(n{\text{/ln}}n)\).

(2) If \(r - s = 1\) for all sufficiently large \(n\), then \({{r}^{3}} = o(n{\text{/l}}{{{\text{n}}}^{3}}n)\).

(3) \(C_{n}^{r}\) and \(C_{{n - s}}^{{r - s}}\) are strictly monotonically increasing functions.

Then there exists \({{k}_{0}}\) such that, for any \(k \geqslant {{k}_{0}}\), the exact equality

$${{R}_{{\max }}}(\{ G(n,r(n), < s(n))\} ,k) = m$$

holds, where m is such that

$$C_{{m - s(m)}}^{{r(m) - s(m)}} < k \leqslant C_{{m + 1 - s(m + 1)}}^{{r(m + 1) - s(m + 1)}}.$$

Obviously, since the function \(C_{{m - s}}^{{r - s}}\) is monotone, the number m in the theorem is uniquely determined. Clearly, an asymptotic representation of m as a function of k can easily be found in many cases. Of much more importance, however, is that the exact value of a Ramsey number has been found for a wide class of parameters. Note that, of course, the classical Ramsey number is not covered by the conditions of the theorem, since \({{K}_{n}} = G(n,1,0) = G(n,1,{\text{ < }}1)\), i.e., the function \(C_{{n - 1}}^{{1 - 1}}\) must grow monotonically, but it is identically equal to unity. Note also that the index \(\max \) is essential here. For the min case, we do not know similar estimates and there is reason to believe that the Ramsey numbers are not well defined in this case.

Theorem 2. Let \({{G}_{n}} = ({{V}_{n}},{{E}_{n}})\), \(n \in \mathbb{N}\), be a sequence of graphs. Suppose that \({{N}_{n}} = {\text{|}}{{V}_{n}}{\text{|}}\) and \({{\alpha }_{n}} = \alpha ({{G}_{n}})\). Let \({{\gamma }_{n}}\) be the maximum number of vertices in Gn that are not adjacent to both vertices of a given edge. Assume that \({{N}_{n}},{{\alpha }_{n}},{{\gamma }_{n}}\) tend monotonically to infinity, where αn = \(o({{N}_{n}})\), and there exists a function \({{\beta }_{n}}\) such that the following asymptotic conditions are satisfied:

(1) \({{\beta }_{n}} > {{\gamma }_{n}}\) and \({{\beta }_{n}} = o({{\alpha }_{n}})\);

(2) \(\mathop {\log }\nolimits_2 {{N}_{n}} = o\left( {\frac{{{{\alpha }_{n}}}}{{{{\beta }_{n}}}}} \right)\);

(3) \(\mathop {\log }\nolimits_2 {{N}_{n}} = o\left( {{{\beta }_{n}} - {{\gamma }_{n}}} \right)\).

Let \({{\varphi }_{n}}\) be an arbitrary function that tends to zero as \(n \to \infty \) and such that \(\mathop {\log }\nolimits_2 {{N}_{n}} = o\left( {{{\varphi }_{n}}({{\beta }_{n}} - {{\gamma }_{n}})} \right)\). Then there exists \({{k}_{0}}\) such that, for any \(k \geqslant {{k}_{0}}\), it is true that \({{R}_{{\max }}}(\{ {{G}_{n}}\} ,k) \geqslant m,\) where m is the maximum positive integer satisfying the conditions \(k \geqslant {{\alpha }_{m}}(1 + {{\varphi }_{m}})\) and \(k < {{N}_{m}}\).

If \(m\) is that such \(k \leqslant {{\alpha }_{m}}\), then

$${{R}_{{\max }}}(\{ {{G}_{n}}\} ,k) < m.$$

On the one hand, the significance of Theorem 2 is that it provides a fairly general result. Specifically, its conditions are satisfied by numerous sequences of graphs from Theorem 1. The fact is that, for them, \({{\alpha }_{n}} = C_{{n - s(n)}}^{{r(n) - s(n)}}\) (see [18, 19]) and \({{N}_{n}} = C_{n}^{{r(n)}}\). For computational simplicity, assume that r is fixed and s = 1 (Kneser graph). Then \({{\gamma }_{n}} \leqslant {{r}^{2}}C_{n}^{{r - 2}}\). This quantity has the order of growth \({{n}^{{r - 2}}}\), while αn grows as \({{n}^{{r - 1}}}\). Thus, the necessary βn is easy to choose. For other parameters, the calculation is slightly more difficult, but the situation is similar. There is also a variety of other sequences of graphs satisfying the conditions of Theorem 2 (cf. [21]).

On the other hand, Theorem 1 gives the exact value of a Ramsey number, while Theorem 2 deals, as a rule, with its asymptotics (which nevertheless is also much stronger than the results on the classical number \(R(k)\)).

It can be seen that the key difference of Theorems 1 and 2 from what we know about the classical number \(R(k)\) is that, for the latter, the independence number of the original graph \({{G}_{n}} = {{K}_{n}}\) is equal to 1, while, in Theorems 1 and 2, it strictly monotonically increases by assumption. In the next section, we describe proof sketches, from which this difference will become even clearer.

3 IDEAS OF THE PROOFS

The upper bounds in both theorems are fairly simple. The matter is that independent sets of vertices are, of course, induced subgraphs in any graph. How this fact can be used is seen directly from the formulation of Theorem 2. In Theorem 1, it is sufficient to use the inequality \(k \leqslant C_{{m + 1 - s(m + 1)}}^{{r(m + 1) - s(m + 1)}}\), the monotonicity of the function on the right-hand side of this inequality, and the fact that this function expresses the independence number of the graph from Theorem 1 (see [18–20]).

The lower bounds are much more complicated. To obtain a lower bound, we need to prove that, for a given m, the initial graph has a spanning subgraph such that each k-vertex subset does not contain at least one edge of the initial graph and contains at least one such edge (as a result, neither the graph nor its complement contain an induced subgraph of the initial graph). We apply a probabilistic method, i.e., consider a random subgraph of the initial graph in which each edge is preserved irrespective of the other edges with probability 1/2. It is proved that, for sufficiently large m, the probability of a set of graphs having the required property is positive, which implies the existence of such graphs. In the proof of Theorem 1, the bounds are derived by applying the methods from [18], while, in the proof of Theorem 2, we use the technique from [21]. An important role is also played by bounds in the spirit of those obtained in [10, 11, 22].

More specifically, it is proved in [18, 21] that, with probability tending to unity, the independence number of a random subgraph of the initial graph is at most \(C_{{m - s(m)}}^{{r(m) - s(m)}}\) in the case of [18] and is strictly less than \({{\alpha }_{m}}(1 + {{\varphi }_{m}})\) in the case of [21]. These results are equivalent to the following one: for all large \(m\), each vertex set of cardinality k contains at least one edge of the initial graph with probability higher than 1/2. However, edges are preserved and deleted with the same probability of 1/2. Therefore, each vertex set of cardinality k does not contain at least one edge of the initial graph with probability higher than 1/2. Hence, with positive probability, there is no vertex set of size k that contains all edges of the initial graph (in which case its induced subgraph would remain) and there is no vertex set of size k that does not contain any edge of the initial graph (in which case its induced subgraph in the complement would remain). Thus, we have established the existence of the sought spanning subgraphs and have proved Theorems 1 and 2.

Note that, in fact, we used the stability of the independence number of a random subgraph. This property has been addressed in numerous works in recent years (see, e.g., [23–25]).