Infinite Combinatorics revisited in the absence of Axiom of Choice

Abstract: 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.