No common hole can be forced over a prescribed $(k-2)$-core (Erdős 624)

Theorem. Let $k\ge3$, $|U|=2^k$, and let $W\subseteq U$ be any set with $|W|=k-2$. Then there are two maps $g_0,g_1:\mathcal P(U)\to U$ such that for every $k$-set $Y\supseteq W$,

Explicitly, $g_i(A)=\theta(i,[A\not\subseteq W],A\cap W)$ for a bijection $\theta:\{0,1\}^2\times\mathcal P(W)\to U$. Context: Erdős Problem 624 asks whether $H(n)-\log_2 n\to\infty$, where $H(n)$ is the least $m$ such that some $f:\mathcal P(X)\to X$ ($|X|=n$) satisfies $f[\mathcal P(Y)]=X$ for all $|Y|\ge m$. By an earlier track-amplification result, it would suffice to show that for each fixed number $r$ of maps $g_i:\mathcal P(U)\to U$, $|U|=2^k$, $k$ large, some $k$-set $Y$ and some $u\in U$ are missed by all $g_i$ on $\mathcal P(Y)$ (a common hole). This shows that the one-map palette argument over a fixed core $W$ (which proves the robust Alon bound) cannot give a two-map common hole: any successful argument must compare different cores.

Formalization Note. $U=\mathrm{Fin}\,(2^k)$.