No l Grid-Points in Spaces of Small Dimension

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Motivated by a question raised by Pór and Wood in connection with compact embeddings of graphs in ${\mathbb Z}^d$ , we investigate generalizations of the no-three-in-line-problem. For several pairs (k,l) we give algorithmic lower, and upper bounds on the largest sizes of subsets S of grid-points from the d-dimensional T × ⋯ ×T-grid, where no l distinct grid-points of S are contained in a k-dimensional affine or linear subspace.