A constructive approach for the lower bounds on the Ramsey numbersR (s, t) (original) (raw)

An upper bound for the Ramsey numbers r(K3,G)

Discrete Mathematics, 1994

The Ramsey number r(H, G) is defined as the minimum N such that for any coloring of the edges of the N-vertex complete graph K N in red and blue, it must contain either a red H or a blue G. In this paper we show that for any graph G without isolated vertices, r(K 3 , G) ≤ 2q + 1 where G has q edges. In other words, any graph on 2q + 1 vertices with independence number at most 2 contains every (isolate-free) graph on q edges. This establishes a 1980 conjecture of Harary. The result is best possible as a function of q.

Upper Bounds of Ramsey Numbers

Applied Mathematical Sciences

The Ramsey number R(G1,G2,…,Gk) is the least integer p so that for any k-edge coloring of the complete graph Kp, there is a monochromatic copy of Gi of color i. In this paper, we derive upper bounds of R(G1,G2,…,Gk) for certain graphs Gi. In particular, these bounds show that R(9,9)⩽6588 and R(10,10)⩽23556 improving the previous best bounds of 6625 and 23854.

New Computational Upper Bounds for Ramsey Numbers R (3, k)

2012

Using computational techniques we derive six new upper bounds on the classical twocolor Ramsey numbers: R(3, 10) ≤ 42, R(3, 11) ≤ 50, R(3, 13) ≤ 68, R(3, 14) ≤ 77, R(3, 15) ≤ 87, and R(3, 16) ≤ 98. All of them are improvements by one over the previously best published bounds.

Constructive lower bounds for Ramsey numbers from linear graphs

Australasian Journal of Combinatorics, 2017

Giraud (1968) demonstrated a process for constructing cyclic Ramsey graph colourings, starting from a known cyclic 'prototype' colouring, adding edges of a single new colour, and producing a larger cyclic pattern. This paper describes an extension of that construction which allows any number of new colours to be introduced simultaneously, by using two multicolour prototypes, each of which is a linear Ramsey graph. The resulting colouring is also linear, which allows the process to be applied iteratively. It is then proved that a simple formula resulting from the new construction provides improved lower bounds for many Ramsey numbers. Giraud's recursive formula is proved for all linear cases, as a corollary. The formula resulting from the new construction is applied to produce new lower bounds for several particular Ramsey numbers, including R 5 (4) ≥ 4176, R 4 (5) ≥ 3282, R 5 (5) ≥ 33495 and R 4 (6) ≥ 20202. For some larger R r (3), the construction produces new lower bounds that improve over the construction described by Chung (1973)-including R 12 (3) ≥ 575666. The paper goes on to explore the general limits, implied by the formula , for lower bounds for the Ramsey numbers R r (k). Specific lower bounds are derived in the form lim r→∞ R r (k) 1/r ≥ g k .

Lower bounds for lower Ramsey numbers

Journal of Graph Theory, 1990

For any graph G, let i(G) and p(G) denote the smallest number of vertices in a maximal independent set and maximal clique, respectively. For positive integers rn and n, the lower Ramsey number s(rn,n) is the largest integer p so that every graph of order p has i(G) I rn or p(G) I n. In this paper we give several new lower bounds for s (rn,n) as well as determine precisely the values s(1,n).

More Constructive Lower Bounds on Classical Ramsey Numbers

SIAM Journal on Discrete Mathematics, 2011

We present several new constructive lower bounds for classical Ramsey numbers. In particular, the inequality R(k, s+1) ≥ R(k, s) + 2k − 2 is proved for k ≥ 5. The general construction permits us to prove that for all integers k, l, with k ≥ 5 and l ≥ 3, the connectivity of any Ramsey-critical (k, l)-graph is at least k, and if k ≥ l − 1 ≥ 1, k ≥ 3 and (k, l) = (3, 2), then such graphs are Hamiltonian. New concrete lower bounds for Ramsey numbers are obtained, some with the help of computer algorithms, including:

Upper Bounds on the Large Ramsey Number LR 2 ( k ) Exposition

Paris and Harrington [2] proved The Large Ramsey Theorem. We will examine the case of Large Ramsey for graphs (stated below). The general case is interesting in that the associated function grows so quickly that the proof is not in Peano Arithmetic. This was Paris and Harrington's motivation (see Appendix). We give upper bounds in the case of 2-coloring the edges of a graph. The graph case is provable in Peano Arithmetic. None of this manuscript is original. not the complete bipartite graph with k vertices on the left and n vertices on the left even though the notation looks similar.) 2. Let K ω be the complete graph on the vertices of N. 3. Let K [k,ω) be the complete graph on the vertices {k, k + 1,. . .}. 4. We will only be coloring EDGES of complete graphs. Henceforth in this manuscript the term coloring G will mean coloring the edges of G. 5. Assume that a complete graph (on a finite or infinite number of vertices) is colored. A homogeneous set is a set of vertices of the g...

Some Graph Theoretic Results Associated with Ramsey's Theorem

We consider the numbers associated with Ramsey's theorem as it pertains to partitions of the pairs of elements of a set into two classes. Our purpose is to give a unified development of enumerative techniques which give sharp upper bounds on these numbers and to give constructive methods for partitions to determine lower bounds on these numbers. Explicit computations include the values of R(3, 6) and R(3, 7) among others. Our computational techniques yield the upper bound R(x, y) < cyX-llog log y/log y for x ~> 3.

A new upper bound on the Ramsey number R(5, 5)

Australas. J Comb., 1992

We show that, in any colouring of the edges of K53 with two colours, there exists a monochromatic K 5 , and hence R(5, 5) ~ 53. This is accomplished in three stages: a full enumeration of (4,4)-good graphs, a derivation of some upper bounds for the maximum number of edges in (4,5)-good graphs, and a proof of the nonexistence of (5,5)-good graphs on 53 vertices. Only the first stage required extensive help from the computer.