  We investigate the provability of classical combinatorial theorems in ZF.
Using combinatorial arguments, we establish the following results for each
infinite cardinal ${\kappa}\in On$,
  (1) ${\kappa}^+\to ({\kappa},{\omega}+1)$,
  (2) any family $\mathcal A\subset [{On}]^{<{\omega}}$ of size ${\kappa}^+$
contains a $\Delta$-system of size ${\kappa}$,
  (3) given a set mapping $F:{\kappa}\to {[{\kappa}]}^{<{\omega}}$, the set
${\kappa}$ has a partition into ${\omega}$-many $F$-free sets,
  By employing Karagila's method of absoluteness, we prove the following for
each uncountable cardinal ${\kappa}\in On$,
  (4) given a set mapping $F:{\kappa}\to {[{\kappa}^]}^{<{\omega}}$, there is
an $F$-free set of cardinality ${\kappa}$,
  (5) for each natural number $n$, every family $\mathcal A\subset
{[{\kappa}]}^{{\omega}}$with $|A\cap B|\le n$ for $\{A,B\}\in {[\mathcal
A]}^{2}$ has property $B$,
  In contrast to (5), we show that the following statement is not provable from
ZF + $cf({\omega}_1)={\omega}_1$:
  (6*) every family $\mathcal A\subset {[{\omega}_1]}^{{\omega}}$ with $|A\cap
B|\le 1$ for $\{A,B\}\in {[\mathcal A]}^{2}$ is "essentially disjoint" .
  The following statements are not provable in ZF, but they are equivalent in
ZF:
  (i) $cf({\omega}_1)={\omega}_1$,
  (ii) ${\omega}_1\to ({\omega}_1,{\omega}+1)^2$,
  (iii) any family $\mathcal A\subset [{On}]^{<{\omega}}$ of size ${\omega}_1$
contains a $\Delta$-system of size ${\omega}_1$.
  A function $f$ is a "uniform denumeration on ${\omega}_1$" iff
$dom(f)={\omega}_1$ and for every ${\alpha}<{\omega}_1$, $f({\alpha})$ is a
function from ${\omega}$ onto ${\alpha}$. It is evident that the existence of a
uniform denumeration of ${\omega}_1$ implies $cf({\omega}_1)={\omega}_1$. We
prove that the failure of the reverse implication is equiconsistent with the
existence of an inaccessible cardinal.
