Abstract. For an oriented graph [Formula: see text] and a set [Formula: see text], the inversion of [Formula: see text] in [Formula: see text] is the digraph obtained by reversing the orientations of the edges of [Formula: see text] with both endpoints in [Formula: see text]. The inversion number of [Formula: see text], [Formula: see text], is the minimum number of inversions which can be applied in turn to [Formula: see text] to produce an acyclic digraph. Answering a recent question of Bang-Jensen, da Silva, and Havet we show that, for each [Formula: see text] and tournament [Formula: see text], the problem of deciding whether [Formula: see text] is solvable in time [Formula: see text], which is tight for all [Formula: see text]. In particular, the problem is fixed-parameter tractable when parameterized by [Formula: see text]. On the other hand, we build on their work to prove their conjecture that for [Formula: see text] the problem of deciding whether a general oriented graph [Formula: see text] has [Formula: see text] is NP-complete. We also construct oriented graphs with inversion number equal to twice their cycle transversal number, confirming another conjecture of Bang-Jensen, da Silva, and Havet, and we provide a counterexample to their conjecture concerning the inversion number of so-called dijoin digraphs while proving that it holds in certain cases. Finally, we asymptotically solve the natural extremal question in this setting, improving on previous bounds of Belkhechine, Bouaziz, Boudabbous, and Pouzet to show that the maximum inversion number of an [Formula: see text]-vertex tournament is [Formula: see text].

A $k$-uniform hypergraph is $s$-almost intersecting if every edge is disjoint from exactly $s$ other edges. Gerbner et al. [SIAM J. Discrete Math., 26 (2012), pp. 1657--1669] conjectured that for every $k$, and $s>s_0(k)$, every $k$-uniform $s$-almost intersecting hypergraph has at most $(s+1)\binom{2k-2}{k-1}$ edges. We prove a strengthened version of this conjecture and determine the extremal graphs. We also give some related results and conjectures.