It is well known that the trichotomy property of cardinals ($∀κ,λ∈Card\;(κ<λ∨κ=λ∨κ>λ)$$∀κ,λ\in\operatorname{Card}(κ<λ∨κ=λ∨κ>λ)$) is equivalent to the axiom of choice.
D. Feldman and M. Orhon had defined in [1] a generalization of the trichotomy property, $\kappa$-trichotomy property for $\kappa>1$:
$\kappa$-trichotomy property is the statement, every family of size $\kappa$ contains a comparable pair.
They then proceed to show that for any finite $\kappa$, the $\kappa$-trichotomy property implies the axiom of choice.
They then stated some result about other possible $\kappa$.
J. Rodbanjong and P. Vejjajiva in [2] had looked at the dual version of the $\kappa$-trichotomy property, they showed that the dual version still imply AC for finite $\kappa$.
Addon: Apparently the first result was first proven by Tarski in 1964, and the proof can be found in [3], and that the result of the dual version was first proven by Asaf in his MSc thesis, Theorem 3.10
From what I could find, not much is known about implications of $\kappa$-trichotomy-like properties, but what about the other direction?
Are there weak choice principles that are known imply trichotomy-like property? (e.g. "every proper class contains a pair of comparable cardinals")
I will also add that if there is more information about implications of trichotomy-like properties that I missed, then I would love to hear about them. It would be interesting to find a weak choice principle that is equivalent to a weak trichotomy property.
[1] Feldman, D., Orhon, M., & Blass, A. (2008). Generalizing Hartogs’ Trichotomy Theorem (Version 1). arXiv. https://doi.org/10.48550/ARXIV.0804.0673
[2] Rodbanjong, Jaruwat; Vejjajiva, Pimpen, A generalization of the trichotomy principle, Thai J. Math. 17, No. 3, 597-605 (2019). ZBL1477.03199.
[3] Rubin, H., & Rubin, J. E. (1985). Equivalents of the axiom of choice, II (pp. 22-23). Amsterdam: North-Holland.
