On the Existence of a Point Subset with 3 or 6 Interior Points

For any finite planar point set in general position, an interior point of the set is a point of the set such that it is not on the boundary of the convex hull of the set . For any positive integer , let be the smallest integer such that every finite planar point set with no three collinear points and with at least interior points has a subset for which the interior of the convex hull of the set contains exactly or interior points of the set . In this paper, we prove that . 1. Introduction In this paper, we focus on finite planar point sets in general position; that is, no three points are collinear. In 1935, Erd？s and Szekeres [1] posed a problem: for any integer , determine the smallest positive integer such that any finite point set of at least points has a subset of points whose convex hull contains exactly vertices. In 1961, they [2] showed that for all integer and then conjectured that for all integer . In 1974, Bonnice [3] proved that and . In 1970, Kalbfleisch et al. [4] showed that . In 2006, the computer solution for was presented by Szekeres and Peters [5]; that, . In 2001, Avis et al. [6] posed an interior point problem: for any integer , determine the smallest positive integer such that any finite point set of at least points has a subset for which the interior of the convex hull of the set contains exactly points in the set . Moreover, they also showed the results that and . In 1974, Bonnice [3] showed that for all integer . In 2008, Wei and Ding [7] showed that for all integer . Moreover, in 2009, they [8] also showed that . In 2011, Sroysang [9] showed that for all integer . Moreover, in 2012, he [10] also showed that for all integer . In 2001, Avis et al. [6] proved that 3 is the smallest positive integer such that any finite point set of at least 3 interior points has a subset for which the interior of the convex hull of the set contains exactly 3 or 4 points in the set . Moreover, they [11] also proved that 7 is the smallest positive integer such that any finite point set of at least 7 interior points has a subset for which the interior of the convex hull of the set contains exactly 4 or 5 points in the set . In 2009, Wei and Ding [12] showed that any planar point set with 3 vertices and 9 interior points has a subset with 5 or 6 interior points of the set . In 2010, Wei et al. [13] proved that 8 is the smallest positive integer such that any finite point set of at least 8 interior points has a subset for which the interior of the convex hull of the set contains exactly 3 or 5 points in the set . In 2012, Sroysang [14] proved that 7 is