The stable roommate problem asks whether an even number of participants, each with a strict ranking of all other participants, can be paired so that there is no blocking pair. A blocking pair consists of two participants who each prefer the other to their assigned partner. The desired pairing is a perfect matching. Unlike the stable marriage problem, a stable roommate instance need not have a solution (Young and Studebaker 2026).
Let 
be the probability that a solution exists when the
participants have independent, uniformly random strict rankings. Young and Studebaker
(2026) reported
 |
as even 
,
where 
is the gamma function. The leading constant is
approximately 2.234914123. This confirms Mertens's predicted exponent but not his
original numerical prediction for the constant (Young and Studebaker 2026).
Young and Studebaker (2026) report using ChatGPT to develop and check parts of the proof.
