Nearly tight exponents for off-diagonal Ramsey numbers

We construct a new family of $K_s$-free graphs that leads to improved lower bounds for Ramsey numbers across a wide range of parameters. For any fixed $s \ge 4$, we show that the off-diagonal Ramsey numbers satisfy $r(s, k) \ge k^{s-2 + o(1)}.$ For $s \ge 6,$ this improves the best known lower bound of the form $r(s, k) \ge k^{\frac{s+1}{2} + o(1)}$ which was first established by Spencer in 1977 and has since only seen logarithmic improvements. This nearly matches the best known upper bound which is of the form $r(s, k) \le k^{s-1 + o(1)}$ and which is widely believed to give the correct exponent. More generally, we show that if $s, k/s \rightarrow \infty$, then $r(s, k) = \left(\frac{k}{s}\right)^{(1+o(1)) s},$ where the upper follows from the seminal work of Erd\H{o}s and Szekeres in 1935. We also obtain improved lower bounds for Ramsey numbers extremely close to the diagonal as well as for diagonal multicolor Ramsey numbers.