The Degree and Codegree Threshold for Generalized Triangle and Some Trees Covering

Given two [Formula: see text]-uniform hypergraphs [Formula: see text] and [Formula: see text], we say that [Formula: see text] has an [Formula: see text]-covering if for every vertex in [Formula: see text] there is a copy of [Formula: see text] cover it. For [Formula: see text], the minimum [Formula: see text]-degree [Formula: see text] of [Formula: see text] is the minimum integer such that every [Formula: see text] vertices are contained in at least [Formula: see text] edges. Let [Formula: see text] be the largest minimum [Formula: see text]-degree among all [Formula: see text]-vertex [Formula: see text]-uniform hypergraphs that have no [Formula: see text]-covering. In this paper, we mainly consider the [Formula: see text]-covering problem in [Formula: see text]-uniform hypergraphs. When [Formula: see text] is a generalized triangle [Formula: see text], we give the exact value of [Formula: see text] and asymptotically determine [Formula: see text]. Moreover, when [Formula: see text] is a linear [Formula: see text]-path [Formula: see text] or a star [Formula: see text], we provide bounds of [Formula: see text] and [Formula: see text] for [Formula: see text], where [Formula: see text].