Even-packet capacity: $d+1$ packets whose zero-sum subsets are even packet unions bound the colours by $b(2^d-1)$ (Erdős 624)

Setting: $U$ is a finite set of inputs, $L:\mathcal P(U)\to\mathcal P(C)$ an arbitrary list map (the actual colour lists), $k$ a natural number. A colour $u$ is nonbottom if $u\notin L(\varnothing)$ and covering if every $k$-subset $Y\subseteq U$ contains some $A$ with $u\in L(A)$. An actual minimal witness of $u$ is $F\subseteq U$ with $u\in L(F)$ and $u\notin L(F')$ for every $F'\subsetneq F$. A valid label for $u$ is a map $p:U\to V$ into an $\mathbb F_2$-vector space with $\sum_{x\in F}p(x)=0$ for every actual minimal witness $F$ with $1\le|F|\le k$. For $T\subseteq U$, $\operatorname{rk}(T)=\dim\operatorname{span}p(T)$ and $\operatorname{rk}(U)=\dim\operatorname{span}p(U)$ (the dimension of the valid parity space). Each colour $u\in S$ has its own valid label $p_u$ into one $\mathbb F_2$-space. Theorem. Assume $|L(A)|\le b$, every $u\in S$ is nonbottom and covering with $\operatorname{rk}_u(U)\ge k-d$ (stated as $k\le\operatorname{rk}_u(U)+d$). Let $P_0,\dots,P_d$ be pairwise disjoint nonempty packets with $|P_0\cup\dots\cup P_d|\le k$ such that for every $u\in S$: $\sum_{P_0\cup P_j}p_u=0$ for $j\ne0$, and every zero-sum $G\subseteq\bigcup_jP_j$ is a union of an even number of packets. Then $|S|\le b(2^d-1)$.

This is the deterministic capacity core of Sol's 'Bias-Free Colour Bounds' (and of 'Bias-Free Bounds on Colour-Dependent Dense Pools'): there, packets with these properties are constructed probabilistically for $s=b(2^d-1)+1$ colours, which this theorem shows is impossible. The probabilistic construction is not formalized.

Context: Erdős Problem 624 asks whether $H(n)-\log_2 n\to\infty$, where $H(n)$ is the least $m$ such that some $f:\mathcal P(X)\to X$ with $|X|=n$ satisfies $f[\mathcal P(Y)]=X$ for all $|Y|\ge m$. This is an auxiliary finite statement from Sol's Round 13 notes on the two-track (PAIR) case. It does not settle Erdős Problem 624, PAIR or Step 17; the initial near-critical population those notes start from is not established.