This interesting paper deals with generalizations of J. H. Silver’s theorem [Ann. Math. Logic 18, 1–28 (1980; Zbl 0517.03018)] stating that any co-analytic equivalence relation with uncountably many equivalence classes has a perfect transversal, and of the Harrington-Marker-Shelah theorem [L. Harrington et al., Trans. Am. Math. Soc. 310, No. 1, 293–302 (1988; Zbl 0707.03042)] which asserts that every Borel partial order can be either written as a countable union of Borel chains or contains a perfect antichain.
To state the main results let us introduce the following notation: A subset \(A\) of a set \(X\) quasi-ordered by a (reflexive and transitive) binary relation \(\leq\) is a chain if every pair of elements of \(A\) is \(\leq\)-comparable, and \(A\) is an antichain if no two elements of \(A\) are \(\leq\)-comparable. A subset \(B\) of a topological space \(Y\) is \(\aleph_0\)-universally Baire if \(\varphi^{-1}[B]\) has the Baire property for every continuous \(\varphi: 2^{\mathbb N}\to Y\). We say that \(B\) is Borel if it is in the \(\sigma\)-algebra generated by all open subsets of \(Y\), \(B\) is analytic if it is a continuous image of the space of the irrationals, and \(B\) is co-analytic if its complement in \(Y\) is analytic.
We can now state the main theorem of the paper:
Theorem 1. Let \(X\) be a Hausdorff topological space and \(\leq\) an \(\aleph_0\)-universally Baire quasi-order such that the orthogonality relation is analytic. Then exactly one of the following holds:

1

The space \(X\) can be written as a union of countably many Borel chains.

2

There is a continuous injection of \(2^\mathbb N\) into an antichain.

Further results deal with possible extensions of the theorem, where the first alternative is weakened to the existence of \(\kappa\)-many \(\kappa^+\)-Borel chains, and present dichotomies for the equivalence relation induced by the quasi-order \(\leq\).
The proofs rely on the combinatorial dichotomies of A. S. Kechris et al. [Adv. Math. 141, No. 1, 1–44 (1999; Zbl 0918.05052)].