On The Cap Set Problem
upper bounds on maximal cardinalities of caps in dimensions seven to ten
N.D. Versluis (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Dion Gijswijt – Mentor
EM van Elderen – Graduation committee member
J.A.M. de Groot – Graduation committee member
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Abstract
This thesis concerns the cap set problem in affine geometry. The problem is illustrated by the card game SET and its geometrical interpretation in ternary affine space. The maximal cardinality of a cap is known for the dimension one to six. For the four lowest dimensions, a maximal cap is constructed and the optimality of its size proven. From there, two recursive methods are described and applied to obtain upper bounds for the maximal size of caps in dimensions seven to ten. The best found upper bounds are 291, 771, 2070 and 5619, respectively.