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The Moore colouring realizes finite binary patterns

Statement

Define $o^{*} (α, β) = * (o (α, β))$ and the binary colouring

$c (α, β) = o (α, β) m o d 2 (α < β < ω_{1}) .$

Let $1 \leq k, l < ω$ , and let $A \subseteq [ω_{1}]^{k}$ and $B \subseteq [ω_{1}]^{l}$ be uncountable pairwise-disjoint families. For every $π : k \to l$ and $χ : k \to 2$ , some $a \in A$ and $b \in B$ satisfy $a < b$ and

$o^{*} (a (i), b (π (i))) = χ (i)$

for all $i < k$ . The same pair has $c (a (i), b (π (i))) = 1 - χ (i)$ ; consequently $c$ realizes every prescribed binary function on the graph ${(i, π (i)) : i < k}$ .

This is a functional-coordinate pattern. It does not assert simultaneous realization of an arbitrary binary matrix on all of $k \times l$ .

Facts & Assumptions

Given: ZFC, positive $k, l$ , uncountable pairwise-disjoint $A, B$ , and maps $π : k \to l$ , $χ : k \to 2$ .

[F1]

The oscillation block lemma supplies arbitrarily long common oscillation blocks whose new points all receive one prescribed continuous label $w$ .

[F2]

Oscillation on lower traces and Moore's modular colouring defines $o$ , the least-nondividing-prime transform $*$ , and the evaluated labels.

[F3]

Minimal-walk weights, labelled lower traces, and the functions e-beta supplies the fixed pairwise-distinct Cantor points $⟨ z_{α} : α < ω_{1} ⟩$ used to evaluate the labels.

[F4]

Chinese remainder theorem for a finite pairwise-coprime list: simultaneous residues determine one class modulo the product, and the resulting bijection preserves addition and multiplication solves a finite system of congruences with pairwise-coprime positive moduli, including its empty-list convention.

[F5]

The Axiom of Choice implies that a countable union of countable sets is countable and supports the uncountable thinning used below.

Proof

technique · direct modular coding

1.1

Choose distinct primes $q_{i} > 8$ for $i < k$ . For every $a \in A$ , the finitely many distinct Cantor points $z_{a (i)}$ supplied by F3 have pairwise-disjoint clopen neighborhoods, so some continuous $w_{a} : 2^{ω} \to ω$ satisfies $w_{a} (z_{a (i)}) = q_{i}$ for all $i < k$ . There are only countably many continuous integer-valued maps on Cantor space. By F5, one value $w$ occurs for an uncountable subfamily; replace $A$ by that subfamily.

F2F3F5given

2.1

Put $Q = \prod_{i < k} q_{i}$ and $N = 6 Q$ . Apply [F1] with this $w$ and block length $N$ . It gives $a \in A$ , members $b_{m} \in B$ , and marked points such that, relative to $b_{0}$ , the evaluated label $q_{i} = w (z_{a (i)})$ occurs exactly $m$ additional times in the oscillation set for $(a (i), b_{m} (π (i)))$ , while every other evaluated-label count is unchanged.

F1step 1.1

3.1

For each $i < k$ , let $O_{i} = Osc (a (i), b_{0} (π (i)))$ and $h_{i} = ∣ {ξ \in O_{i} : μ (a (i), b_{0} (π (i)); a (i)) (ξ) = q_{i}} ∣$ . For $s \neq 0, q_{i}$ put $h_{i, s} = ∣ {ξ \in O_{i} : μ (a (i), b_{0} (π (i)); a (i)) (ξ) = s} ∣$ , and set $r_{i} = (\sum_{s \neq 0, q_{i}} (h_{i, s} m o d s)) m o d 6$ . The sum has finite support by F2. The primes $q_{i}$ are pairwise coprime, so F4 gives a residue $x$ modulo $Q$ satisfying $x + h_{i} \equiv 6 - r_{i} + 2^{χ (i)} (m o d q_{i})$ for every $i$ . Choose its representative $0 \leq x < Q < N$ and put $b = b_{x}$ . The right-hand side lies between $2$ and $8$ , hence is already its least nonnegative residue modulo $q_{i}$ .

F2F4step 2.1

4.1

Fix $i < k$ . The block count from step 2.1 and the definition of $r_{i}$ give an integer $y$ such that $o (a (i), b (π (i))) = ((x + h_{i}) m o d q_{i}) + r_{i} + 6 y = 2^{χ (i)} + 6 (y + 1)$ . If $χ (i) = 0$ , this number is odd, so the least prime not dividing it is $2 = p_{0}$ . If $χ (i) = 1$ , it is even but is congruent to $2$ modulo $3$ , so the least prime not dividing it is $3 = p_{1}$ . Thus [F2] gives $o^{*} (a (i), b (π (i))) = χ (i)$ .

F2step 2.1step 3.1

5.1

The same displayed formula is odd exactly when $χ (i) = 0$ , so $c (a (i), b (π (i))) = 1 - χ (i)$ . Given a desired binary pattern $ψ : k \to 2$ for $c$ , apply the proved assertion with $χ = 1 - ψ$ . This proves every claimed functional pattern, including all coordinates at once, and makes no claim about two values in the same row of a nonfunctional matrix.

step 4.1

6.1

Positivity of $k$ is part of the nonvacuous uncountable-family hypothesis; for a formal empty coordinate list F4 would supply the unique empty residue class and the conclusion would be vacuous. Finite prime choice and the least representative require no further choice; the only new AC use is the uncountable thinning in step 1.1.

F4F5step 1.1step 5.1∎

Depends on

Used by

Moore's clopen-generated topology Definition
An oscillation pattern controls clopen membership Example
Moore's topology is hereditarily Lindelof Lemma
The Moore colouring forbids cross-injections Lemma

Dependency tree · two levels

23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Moore, A solution to the L space problem, Section 5, Theorem 5.3 and proof, printed pp. 15–16 (standard reference, not scraped)