We introduce compactness theorems for generalized 
colorings and derive several particular compactness 
theorems from them. 
It is proved that the theorems and many of 
their consequences are equivalent in ZF set theory to BPI, 
the Prime Ideal Theorem for Boolean algebras. \\
\noindent {\bf keywords:} generalized graph colorings, 
compactness, prime ideal theorem \\
\noindent {\em MSC:} 05C15; 03E25 .