Triangle-free matchstick graphs on n vertices have at most 2n minus a square-root term edges, and matchstick graphs in a disk of radius r are claimed to have at most (2-ε(r))n edges.
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On the number of edges of restricted matchstick graphs
Triangle-free matchstick graphs on n vertices have at most 2n minus a square-root term edges, and matchstick graphs in a disk of radius r are claimed to have at most (2-ε(r))n edges.