Abstract: Whether the successor of a singular cardinal can be Jónsson is a very old and famous open problem in set theory.
Here, we collect necessary conditions for an affirmative answer, and put forward a list of closely-related questions in singular cardinals combinatorics that could eventually lead to settling the main problem at hand.
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This piece was originally written in 2023 for the proceedings of the Perspectives on Set Theory conference, but that special volume got canceled. It was updated in November 2025 and was accepted for publication at the Bollettino dell’Unione Matematica Italiana on December 2025.