  In 1962 P\'osa conjectured that every graph G on n vertices with minimum
degree at least 2n/3 contains the square of a hamiltonian cycle. In 1996 Fan
and Kierstead proved the path version of P\'osa's Conjecture. They also proved
that it would suffice to show that G contains the square of a cycle of length
greater than 2n/3. Still in 1996, Koml\'os, S\'ark\"ozy, and Szemer\'edi proved
P\'osa's Conjecture, using the Regularity and Blow-up Lemmas, for graphs of
order n > n_0, where n_0 is a very large constant. Here we show without using
these lemmas that n_0=2\times 10^8 is sufficient. We are motivated by the
recent work of Levitt, Szemer\'edi and S\'ark\"ozy, but our methods are based
on techniques that were available in the 90's.

Combinatorics