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

IAENG International Journal of Applied Mathematics

For positive integers k and l , the Ramsey number R(k,l) is the least positive integer n such that for every graph G of order n , either G contains k K as a subgraph or G contains l K as a subgraph. In this paper it is shown that Ramsey numbers ≥ R(k,l) 2kl -3k -3l + 6 when ≤ ≤ 3 k l , and ≥ R(k,l) 2kl -3k + 2l -12 when ≤ ≤ 5 k l .

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 .

Two extensions of Ramsey’s theorem

Duke Mathematical Journal, 2013

Ramsey's theorem, in the version of Erdős and Szekeres, states that every 2-coloring of the edges of the complete graph on {1, 2,. .. , n} contains a monochromatic clique of order 1 2 log n. In this paper, we consider two well-studied extensions of Ramsey's theorem. Improving a result of Rödl, we show that there is a constant c > 0 such that every 2-coloring of the edges of the complete graph on {2, 3, ..., n} contains a monochromatic clique S for which the sum of 1/ log i over all vertices i ∈ S is at least c log log log n. This is tight up to the constant factor c and answers a question of Erdős from 1981. Motivated by a problem in model theory, Väänänen asked whether for every k there is an n such that the following holds. For every permutation π of 1,. .. , k − 1, every 2-coloring of the edges of the complete graph on {1, 2,. .. , n} contains a monochromatic clique a 1 <. .. < a k with a π(1)+1 − a π(1) > a π(2)+1 − a π(2) >. .. > a π(k−1)+1 − a π(k−1). That is, not only do we want a monochromatic clique, but the differences between consecutive vertices must satisfy a prescribed order. Alon and, independently, Erdős, Hajnal and Pach answered this question affirmatively. Alon further conjectured that the true growth rate should be exponential in k. We make progress towards this conjecture, obtaining an upper bound on n which is exponential in a power of k. This improves a result of Shelah, who showed that n is at most double-exponential in k.

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 further results in Ramsey graph construction

Australiasian Journal of Combinatorics, 2020

A construction described by the current author (2017) uses two linear 'prototype' graphs to build a compound graph with Ramsey properties inherited from the prototypes. The resulting graph is linear; and cyclic if both prototypes are cyclic. However, it will not generate a cyclic graph from a general linear prototype. Building on the properties of that construction, this paper proves that a general linear prototype graph of order m can be extended using a single new colour to produce a new cyclic graph of order 3m − 1 which is triangle-free in the new colour, and has the same clique-number as the prototype in every other colour. The paper then describes a cyclic Ramsey (3, 3, 4, 4; 173)-graph derived by constrained tree search-thus proving that R(3, 3, 4, 4) ≥ 174. Using a quadrupling construction to produce a further cyclic graph, it is shown that R(3, 4, 5, 5) ≥ 693. A compound cyclic Ramsey (3, 7, 7; 622)-graph derived by a limited manual search is then described. Further construction steps produce a (8, 8, 8; 6131)-graph, showing that R 3 (8) ≥ 6132. The paper concludes by showing that R 4 (7) ≥ 81206 and R 4 (9) ≥ 630566, implying corresponding improvements in the lower bounds for R 5 (7) and R 5 (9) and beyond.

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.

On a variation of the Ramsey number

Transactions of the American Mathematical Society, 1972

Let c ( m , n ) c(m,n) be the least integer p p such that, for any graph G G of order p p , either G G has an m m -cycle or its complement G ¯ \bar G has an n n -cycle. Values of c ( m , n ) c(m,n) are established for m , n ⩽ 6 m,n \leqslant 6 and general formulas are proved for c ( 3 , n ) , c ( 4 , n ) c(3,n),c(4,n) , and c ( 5 , n ) c(5,n) .

Graph classes with linear Ramsey numbers

Discrete Mathematics, 2021

The Ramsey number R X (p, q) for a class of graphs X is the minimum n such that every graph in X with at least n vertices has either a clique of size p or an independent set of size q. We say that Ramsey numbers are linear in X if there is a constant k such that R X (p, q) ≤ k(p + q) for all p, q. In the present paper we conjecture that if X is a hereditary class defined by finitely many forbidden induced subgraphs, then Ramsey numbers are linear in X if and only if X excludes a forest, a disjoint union of cliques and their complements. We prove the "only if" part of this conjecture and verify the "if" part for a variety of classes. We also apply the notion of linearity to bipartite Ramsey numbers and reveal a number of similarities and differences between the bipartite and non-bipartite case.

New Ramsey bounds from cyclic graphs of prime order

SIAM Journal on Discrete Mathematics, 1997

We present new explicit lower bounds for some Ramsey numbers. All the graphs are cyclic, and are on a prime number of vertices. We give a partial probabilistic analysis which suggests that the cyclic Ramsey numbers grow exponentially. We show that the standard expectation arguments are insu cient to prove such a result. These arguments motivated our searching for Ramsey graphs of prime order.