On the orchard type problems in higher dimensions
Combinatorics Seminar
26th November 2024, 11:00 am – 12:00 pm
Fry Building, 2.04
Let P be a set of n points in the plane not all on a line. The orchard planting problem asks for the maximum number of the lines through exactly three points of P. Green and Tao showed that the maximum possible number of such lines for an n element set is ⌊n(n − 3)/6⌋ + 1. Lin and Swanepoel also investigated a generalization of the orchard problem in higher dimensions. Namely, if P is a set of n points in d dimension, then they found an upper bound for the maximum number of hyperplanes through exactly d+1 points of P. In this talk we will see that if P is a set which is contained in an algebraic curve C of degree r and determines cn^d hyperplanes through exactly d+1 points of P, then r=d+1, and C is the intersection of [(d+1)(d-2)]/2 quadric hypersurfaces.
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