Random Graphs 93 Proceedings of the sixth international seminar on Random graphs and probabilistic methods in combinatorics and computer science
The Algorithmic Aspects of Uncrowded Hypergraphs
SIAM Journal on Computing
An Algorithm for Heilbronn's Problem
SIAM Journal on Computing
A Lower Bound for Heilbronn's Triangle Problem in d Dimensions
SIAM Journal on Discrete Mathematics
The On-Line Heilbronn's Triangle Problem in Three and Four Dimensions
COCOON '02 Proceedings of the 8th Annual International Conference on Computing and Combinatorics
Distributions of points in the unit-square and large k-gons
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
An Upper Bound for the d-Dimensional Analogue of Heilbronn's Triangle Problem
SIAM Journal on Discrete Mathematics
Distributions of points in d dimensions and large k-point simplices
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
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For fixed integers k茂戮驴 3 and hypergraphs ${\mathcal G}$ on Nvertices, 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(Ti茂戮驴 1·(ln T)(k茂戮驴 i)/(k茂戮驴 1)), i= 3, ..., k, for some number T茂戮驴 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 Iof size |I| = 茂戮驴((N/T) ·(ln T)1/(k茂戮驴 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 npoints in the unit square [0,1]2.