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 Ramseycritical (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: R(5, 17) ≥ 388, R(5, 19) ≥ 411, R(5, 20) ≥ 424, R(6, 8) ≥ 132, R(6, 12) ≥ 263, R(7, 8) ≥ 217, R(7, 9) ≥ 241, R(7, 12) ≥ 417, R(8, 17) ≥ 961, R(9, 10) ≥ 581, R(12, 12) ≥ 1639, and also one three-color case R(8, 8, 8) ≥ 6079. [PUBLICATION ABSTRACT]