Asymptotic Sharpness of Bounds on Hypertrees

The hypertree can be defined in many different ways. Katona and Szabó introduced a new, natural definition of hypertrees in uniform hypergraphs and investigated bounds on the number of edges of the hypertrees. They showed that a $k$-uniform hypertree on $n$ vertices has at most \( \binom{n}{k−1} \) edges and they conjectured that the upper bound is asymptotically sharp. Recently, Szabó verified that the conjecture holds by recursively constructing an infinite sequence of $k$-uniform hypertrees and making complicated analyses for it. In this note we give a short proof of the conjecture by directly constructing a sequence of $k$-uniform $k$-hypertrees.