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Bounds on the Number of Edges of Edge-Minimal, Edge-Maximal and L-Hypertrees

Abstract In their paper, Bounds on the number of edges in hypertrees, G.Y. Katona and P.G.N. Szabó introduced a new, natural definition of hypertrees in k- uniform hypergraphs and gave lower and upper bounds on the number of edges. They also defined edge-minimal, edge-maximal and l-hypertrees and proved an upper bound on the edge number of l-hypertrees. In the present paper, we verify the asymptotic sharpness of the upper bound on the number of edges of k-uniform hypertrees given in the above mentioned paper. We also make an improvement on the upper bound of the edge number of 2-hypertrees and give a general extension construction with its consequences. We give lower and upper bounds on the maximal number of edges of k-uniform edge-minimal hypertrees and a lower bound on the number of edges of k-uniform edge-maximal hypertrees. In the former case, the sharp upper bound is conjectured to be asymptotically .