Conditioned Alon bound: Alon-type loss on every star of $k$-sets through a small core (Erdős 624)

Let $U$ be a finite set with $n=2^k$ elements. For a map $g:\mathcal P(U)\to U$ and $Y\subseteq U$ write $I_g(Y)=\{g(A):A\subseteq Y\}$. For $B\subseteq U$ let $\mathcal K_B$ be the set of $k$-subsets $Y\supseteq B$.

Theorem. Let $k\ge3$, $B\subseteq U$ with $|B|\le k-2$, and let $g:\mathcal P(U)\to U$ be arbitrary. Then

So on average over the $k$-sets containing any prescribed core of size at most $k-2$, every map misses at least $n/32$ points. With $B=\emptyset$ this is the earlier robust_alon_sum; the conditioned version is strictly stronger and does not follow from the unconditioned average. 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 large $k$, any two maps $\mathcal P(U)\to U$ with $|U|=2^k$ have a common hole: a $k$-set $Y$ and a point $u$ missed by both maps on $\mathcal P(Y)$. These results do not settle Problem 624.

Formalization Note. Counting form with natural numbers (truncated subtraction is harmless since $|I_g(Y)|\le n$); $U$ is an arbitrary Fintype with Fintype.card U = 2 ^ k.