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.