  Given positive integers $m_1, m_2, ..., m_n$, and $n$ general points $p_i$ of
${\bf CP}^2$, bounds are given for the least degree $t$ among plane curves
passing through each point $p_i$ with multiplicity at least $m_i$, and for the
least $t$ such that the $n$ multiple points impose independent conditions on
curves of degree $t$, often improving substantially what was previously known.
As an application, the Hilbert function (resp., minimal free resolution) is
determined for symbolic powers $I^{(m)}$ for the ideal $I$ defining $n$ general
points of ${\bf CP}^2$ for infinitely many m for each square n (resp., for
infinitely many m for each even square n). Four graphs are included showing
other values of m and n for which results are given.
