Abstract. We give, for each [Formula: see text], the precise best possible minimum positive codegree condition for a perfect matching in a large [Formula: see text]-uniform hypergraph [Formula: see text] on [Formula: see text] vertices. Specifically, we show that if [Formula: see text] is sufficiently large and divisible by [Formula: see text] and [Formula: see text] has minimum positive codegree [Formula: see text] and no isolated vertices, then [Formula: see text] contains a perfect matching. For [Formula: see text], this was previously established by Halfpap and Magnan [ Positive Co-Degree Thresholds for Spanning Structures, 2024], who also gave bounds for [Formula: see text] which were tight up to an additive constant.

Abstract. We study conditions under which an edge-colored hypergraph has a particular substructure that contains more than the trivially guaranteed number of monochromatic edges. Our main result solves this problem for perfect matchings under minimum degree conditions. This answers recent questions of Gishboliner, Glock, and Sgueglia and of Balogh, Treglown, and Zárate-Guerén.

Abstract. A result of Balogh et al. yields the minimum degree threshold that ensures a 2-colored graph contains a perfect matching of significant color-bias (i.e., a perfect matching that contains significantly more than half of its edges in one color). In this note we prove an analogous result for perfect matchings in [Formula: see text]-uniform hypergraphs. More precisely, for each [Formula: see text] and [Formula: see text] we determine the minimum [Formula: see text]-degree threshold for forcing a perfect matching of significant color-bias in an [Formula: see text]-colored [Formula: see text]-uniform hypergraph.