Abstract
For fixed integers k ≥ 3 and hypergraphs \({\mathcal G}\) on N vertices, which contain edges of cardinalities at most k, and are uncrowded, i.e., do not contain cycles of lengths 2,3, or 4, and with average degree for the i-element edges bounded by O(T i − 1 ·(ln T)(k − i)/(k − 1)), i = 3, ..., k, for some number T ≥ 1, we show that the independence number \(\alpha ({\mathcal G})\) satisfies \(\alpha ({\mathcal G}) = \Omega ((N/T) \cdot (\ln T)^{1/(k-1)})\). Moreover, an independent set I of size |I| = Ω((N/T) ·(ln T)1/(k − 1)) can be found deterministically in polynomial time. This extends a result of Ajtai, Komlós, Pintz, Spencer and Szemerédi for uncrwoded uniform hypergraphs. We apply this result to a variant of Heilbronn’s problem on the minimum area of the convex hull of small sets of points among n points in the unit square [0,1]2.
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Lefmann, H. (2007). Convex Hulls of Point-Sets and Non-uniform Hypergraphs. In: Kao, MY., Li, XY. (eds) Algorithmic Aspects in Information and Management. AAIM 2007. Lecture Notes in Computer Science, vol 4508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72870-2_27
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DOI: https://doi.org/10.1007/978-3-540-72870-2_27
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