For integers
, the diagonal
Ramsey number
is the
minimum
such that every
-coloring of the edges of
a complete graph on
vertices yields a monochromatic subgraph on
vertices. Here we make a careful effort of extracting explicit upper bounds for
from the
pigeonhole principle alone. Our main term improves on previously documented explicit
bounds for
,
and we also consider an often-ignored secondary term, which allows us to subtract
a positive proportion of the main term that is uniformly bounded below.
Asymptotically, we give a self-contained proof that
and we conclude by noting that our methods combine with previous estimates on
to improve
the constant
to
,
where
.
We also compare our formulas, and previously documented formulas, to some
collected numerical data.
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