Uniform finite blocks and rapid dilation

---

Source check

Grow–Whicher, Finite unions of quasi-independent sets, printed pp. 491–492, states and proves the uniform-finite reduction, which is restated here as the premise of the block lemma. The argument below is written out for positive integer blocks and uses no analytic Sidon theorem.

The block lemma — proved

Let $E_t\subset\mathbb N_{>0}$ be finite and nonempty. Suppose one $\delta>0$ satisfies $r(F)\geq\delta|F|$ for every $F\subseteq E_t$ and every $t$. Choose positive integers $p_t$ with $p_1=1$ and

$$p_{t+1}>\sum_{i\leq t}p_i\sum_{x\in E_i}x. \tag{1}$$

Put $A=\bigcup_t p_tE_t$. The blocks are pairwise disjoint: the smallest element of block $t+1$ exceeds the sum of all earlier block elements. In particular $A$ is infinite.

Consider any finite signed relation in $A$, and write it as

$$\sum_{i=1}^m p_i u_i=0, \qquad u_i=\sum_{x\in E_i}\epsilon_{i,x}x\in\mathbb Z, \quad\epsilon_{i,x}\in\{-1,0,1\}.$$

If some $u_i\neq0$, choose the largest such index $j$. If $j=1$, the relation already contradicts $p_1u_1\neq0$. If $j>1$, then

$$|p_ju_j|\geq p_j> \sum_{i<j}p_i\sum_{x\in E_i}x\geq \left|\sum_{i<j}p_i u_i\right|,$$

again a contradiction. Hence every $u_i=0$. This separates all signed relations, without a bound on their lengths.

For finite $B\subset A$, extract a dissociated subset from each of its block intersections with relative size at least $\delta$. Their union is dissociated by the preceding separation argument and has size at least $\delta|B|$.

If also $\chi(E_t)>t$, a partition of $A$ into $q$ dissociated classes would restrict and rescale to such a partition of $E_q$, a contradiction. Thus this extra hypothesis gives the required counterexample. The extra hypothesis has not been achieved.

Finite determination — proved

For a countable integer set $A$ and fixed $q$, if every finite subset admits a dissociated $q$-coloring, so does $A$. Enumerate $A$. Consider the finitely branching tree whose level $n$ is the set of valid $q$-colorings of its first $n$ elements, with restriction as the parent map. Every level is nonempty. Repeatedly choose a child with arbitrarily deep descendants; this is possible because the number of children is finite. The resulting branch colors $A$. Any relation has finite support and therefore would occur at one level, so every branch color class is dissociated.

Consequently an infinite counterexample with extraction constant $\delta$ would itself supply finite witnesses $E_q$ with the same constant and $\chi(E_q)>q$. The finite-block target is therefore exact.