arXiv:2011.00915v1 [math.CO] 2 Nov 2020 At most 4.47 n stable matchings Cory Palmer * D¨om¨ot¨ or P´ alv¨ olgyi † November 3, 2020 Abstract We improve the upper bound for the maximum possible number of stable matchings among n men and n women from O(131072 n ) to O(4.47 n ). To establish this bound, we develop a novel formulation of a probabilistic technique that is easy to apply and may be of independent interest in counting other combinatorial objects. 1 Introduction Since their introduction in 1962 by Gale and Shapley [8], stable matchings have found a wide variety of practical applications ranging from the classic college admissions through hospital residences to life-saving kidney exchanges [4, 5]. Its importance is shown by the 2012 Nobel Memorial Prize in Economic Sciences awarded to Shapley and Roth for their work on the topic. An important question when designing algorithms is the maximum possible number of stable matchings among n men and n women. This has been proposed to study by Knuth in 1976 [14]; see also Manlove [15]. Despite the considerable attention the question has received, a simply exponential upper bound has been obtained only recently by Karlin, Gharan and Weber [11]. They claimed that their bound is about 2 17n and could see no way of substantially improving it to get close to the best known lower bound Ω(2.28 n ) of Irving and Leather [10]. The importance to give a better upper bound was also highlighted in the recent survey of Cechl´ arov´a, Cseh and Manlove [7]. Our main result is to improve the upper bound for the maximum possible number of stable matchings to O(4.47 n ). Apart from its theoretical importance, this also improves the known upper bound for the running time of several algorithms; for example, the upper bound of the running time of the best known algorithm for the popular roommates problem with strict preferences by Kavitha [13] is improved from roughly 400000 n to 13.41 n . Our starting point will be the same as in [11], for which we need the following definition, implicitly contained in their work. A poset P is called an n × n tangled grid if it can be covered by n chains, M 1 ,...,M n , and also by n other chains, W 1 ,...,W n , which have the additional property that |M i ∩ W j |≤ 1 for any i, j . Note that no restrictions are posed on the lengths of the chains but because we require both M 1 ,...,M n and W 1 ,...,W n to cover P , for each chain |M i |≤ n and |W j |≤ n. Also note that M i and M j are not required to be disjoint. Because of these P might have less than n 2 elements. In * Department of Mathematical Sciences, University of Montana, Missoula, Montana 59812, USA. E-mail: cory.palmer@umontana.edu. † MTA-ELTE Lend¨ ulet Combinatorial Geometry Research Group, Institute of Mathematics, E¨ otv¨os Lor´ and Uni- versity (ELTE), Budapest, Hungary. E-mail: dom@cs.elte.hu. 1