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In Sturmfels and Whiteley (J Symb Comput 11(5):439-453, ), it is proven that any multihomogenous bracket polynomial with integer coefficients can be interpreted by a ruler construction when introducing simple non-degeneracy conditions. This problem is called (generalized) Cayley factorization. This allows for geometrically interpreting many projective invariant properties. We reprove the above statement in rank 3 giving a better bound on the size of the non-degeneracy conditions. The constant factor in the bound is essentially reduced from 105 to 9. The algorithm described is concise enough to be implemented on a computer. With this algorithm, interpreting a common condition for six points to lie on a common conic is very close to Pascal's construction (see Apel, in: The geometry of brackets and the area principle, Dissertation, TU München, ). [ABSTRACT FROM AUTHOR]

Abstract   Erdős asked whether every sufficiently large set of points in general position in the plane contains six points that form a convex hexagon without any points from the set in its interior. Such a configuration is called an empty convex hexagon. In this paper, we answer the question in the affirmative. We show that every set that contains the vertex set of a convex 9-gon also contains an empty convex hexagon. [ABSTRACT FROM AUTHOR]