Bounding 2d functions by products of 1d functions

Abstract

Given sets X,Y$X,Y$ and a regular cardinal μ, let normalΦfalse(X,Y,μfalse)$\Phi (X,Y,\mu )$ be the statement that for any function f:X×Y→μ$f : X \times Y \rightarrow \mu$, there are functions g1:X→μ$g_1 : X \rightarrow \mu$ and g2:Y→μ$g_2 : Y \rightarrow \mu$ such that for all false(x,yfalse)∈X×Y$(x,y) \in X \times Y$, f(x,y)≤maxfalse{g1(x),g2(y)false}$f(x,y) \le \max \lbrace g_1(x), g_2(y) \rbrace$. In ZFC$\mathsf {ZFC}$, the statement normalΦfalse(ω1,ω1,ωfalse)$\Phi (\omega _1, \omega _1, \omega )$ is false. However, we show the theory sans-serifZF+“the4.ptclub4.ptfilter4.pton4.ptω14.ptis4.ptnormal”+normalΦfalse(ω1,ω1,ωfalse)$\mathsf {ZF}+ \text{``the club filter on $\omega _1$ is normal''} + \Phi (\omega _1, \omega _1, \omega )$ (which is implied by sans-serifZF+sans-serifDC$\mathsf {ZF}+ \mathsf {DC}$+ “V=Lfalse(double-struckRfalse)$V = L(\mathbb {R})$” + “ω1 is measurable”) implies that for every α<ω1$\alpha &lt; \omega _1$ there is a κ∈false(α,ω1false)$\kappa \in (\alpha ,\omega _1)$ such that in some inner model, κ is measurable with Mitchell order ≥α$\ge \alpha$.

Maker–breaker Games on And

“…First of all, the club filter on is a countably complete free ultrafilter under ZF+DC+AD (this is explicit in the proof of [9, Theorem 28.2]). Furthermore, it is normal [6, Proposition 4.1]. Thus for any 2-colouring of there exists a colour with a monochromatic (the standard proof of this for arbitrary normal ultrafilters uses only ZF, see [8, Theorem 10.22]).…”

“…The axiom-system ZF+DC+“ is measurable” is equiconsistent with ZFC+“there exists a measurable cardinal” (see [7]). The club filter being an ultrafilter is a strictly stronger assumption; for more details see page 3 of [6].…”