Common-pool low-rank bound (C): $|S|\le b(2^{R+d}-1)$ (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). Here each colour $u$ in a set $S$ has its own valid label $p_u:U\to V$ (all labels taking values in one $\mathbb F_2$-space $V$, which loses no generality). Theorem (C). Suppose every list has size $|L(A)|\le b$, every $u\in S$ is nonbottom and covering with valid label $p_u$ of global rank $\operatorname{rk}_u(U)\ge k-d$ (stated as $k\le\operatorname{rk}_u(U)+d$), and there is one common pool $P\subseteq U$ with $\operatorname{rk}_u(P)\le R$ for all $u\in S$, where $R+d\le k$ and $R+d\le|P|$. Then $|S|\le b\,(2^{R+d}-1)$.

Source: 'Individual Spectra and Common-Input Degree', boxed bound (C); it also contains the equal-label capacity bound of 'Coordinate-Label Diversity and Low-Rank Tails' as a special case.

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.