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Motivated by a question of Grinblat, we study the minimal number v ( n ) that satisfies the following. If A 1 , … , A n are equivalence relations on a set X such that for every i ∈ [ n ] there are at least v ( n ) elements whose equivalence classes with respect to A i are nontrivial, then A 1 , … , A n contain a rainbow matching, i.e. there exist 2 n distinct elements x 1 , y 1 , … , x n , y n ∈ X with x i ∼ A i y i for each i ∈ [ n ] . Grinblat asked whether v ( n ) = 3 n − 2 for every n ≥ 4 . The best-known upper bound was v ( n ) ≤ 16 n / 5 + O ( 1 ) due to Nivasch and Omri. Transferring the problem into the setting of edge-coloured multigraphs, we affirm Grinblat's question asymptotically, i.e. we show that v ( n ) = 3 n + o ( n ) . [ABSTRACT FROM AUTHOR]

In the Oriented-cycle game, introduced by Bollobás and Szabó [7] , two players, called OMaker and OBreaker, alternately direct edges of K n . OMaker directs exactly one previously undirected edge, whereas OBreaker is allowed to direct between one and b previously undirected edges. OMaker wins if the final tournament contains a directed cycle, otherwise OBreaker wins. Bollobás and Szabó [7] conjectured that for a bias as large as n − 3 OMaker has a winning strategy if OBreaker must take exactly b edges in each round. It was shown recently by Ben-Eliezer, Krivelevich and Sudakov [6] , that OMaker has a winning strategy for this game whenever b < n / 2 − 1 . In this paper, we show that OBreaker has a winning strategy whenever b > 5 n / 6 + 1 . Moreover, in case OBreaker is required to direct exactly b edges in each move, we show that OBreaker wins for b ⩾ 19 n / 20 , provided that n is large enough. This refutes the conjecture by Bollobás and Szabó. [ABSTRACT FROM AUTHOR]