= Finite-injury computable colouring of pairs
There is a two-colour <computable colouring> of pairs with no infinite <computably enumerable set> homogeneous for it. Enumerate these <sets> as $W_e$. In stages, give each eligible requirement two markers from its enumerated <set>, avoiding markers of higher priority, and cancel all lower-priority markers whenever a new assignment is made. Colour $(x,y)$ by the marker status of $x$ at stage $y$: first markers have colour zero, second markers colour one, and unmarked points colour zero. Each requirement is injured only finitely often by induction on its index. If $W_e$ is infinite it eventually has permanent distinct markers $a_e,b_e$; for every sufficiently large $y\in W_e$, the pairs $(a_e,y)$ and $(b_e,y)$ have different colours. Simulating finitely many stages computes each pair's colour, proving both effectiveness and the obstruction.
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