A constructive approach for the lower bounds on the Ramsey numbers R (s, t)

Zarankiewicz Numbers and Bipartite Ramsey Numbers

The Zarankiewicz number z(b; s) is the maximum size of a subgraph of Kb,b which does not contain Ks,s as a subgraph. The two-color bipartite Ramsey number b(s, t) is the smallest integer b such that any coloring of the edges of Kb,b with two colors contains a Ks,s in the rst color or a Kt,t in the second color.In this work, we design and exploit a computational method for bounding and computing...

More Constructive Lower Bounds on Classical Ramsey Numbers

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) 6= (3, 2), then such graphs are Hami...

A Note on Constructive Lower Bounds for the Ramsey Numbers R(3, t)

Constructive lower bounds for Ramsey numbers from linear graphs

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 ...

Constructive Lower Bounds on Classical Multicolor Ramsey Numbers

This paper studies lower bounds for classical multicolor Ramsey numbers, first by giving a short overview of past results, and then by presenting several general constructions establishing new lower bounds for many diagonal and off-diagonal multicolor Ramsey numbers. In particular, we improve several lower bounds for Rk(4) and Rk(5) for some small k, including 415 ≤ R3(5), 634 ≤ R4(4), 2721 ≤ R...