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

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

A note on lower bounds for hypergraph Ramsey numbers

We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that r3(l, l, l) ≥ 2 c log log l . The old bound, due to Erdős and Hajnal, was r3(l, l, l) ≥ 2 2 log2 .

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

Graph G is a (k, p)-graph if G does not contain a complete graph on k vertices Kk, nor an independent set of order p. Given a (k, p)graph G and a (k, q)-graph H, such that G and H contain an induced subgraph isomorphic to some Kk−1-free graph M , we construct a (k, p + q − 1)-graph on n(G) + n(H) + n(M) vertices. This implies that R(k, p+ q − 1) ≥ R(k, p) + R(k, q) + n(M) − 1, where R(s, t) is ...

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...

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...