  In a recent breakthrough Campos, Griffiths, Morris and Sahasrabudhe obtained
the first exponential improvement of the upper bound on the diagonal Ramsey
numbers since 1935. We shorten their proof, replacing the underlying book
algorithm with a simple inductive statement. This modification allows us
  - to give a very short proof of an improved upper bound on the off-diagonal
Ramsey numbers, which extends to the multicolor setting, and
  - to clarify the dependence of the bounds on underlying parameters and
optimize these parameters, obtaining, in particular, an upper bound $$R(k,k)
\leq (3.8)^{k+o(k)}$$ on the diagonal Ramsey numbers.
