Pairwise Balanced Designs of Dimension Three




Niezen, Joanna

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A linear space is a set of points and lines such that any pair of points lie on exactly one line together. This is equivalent to a pairwise balanced design PBD(v, K), where there are v points, lines are regarded as blocks, and K ⊆ Z≥2 denotes the set of allowed block sizes. The dimension of a linear space is the maximum integer d such that any set of d points is contained in a proper subspace. Specifically for K = {3, 4, 5}, we determine which values of v admit PBD(v,K) of dimension at least three for all but a short list of possible exceptions under 50. We also observe that dimension can be reduced via a substitution argument.



pairwise balanced design, PBD, Latin square, MOLS, GDD, dimension, linear space, game of Set, Steiner triple system, STS, Steiner space, subspace, Wilson's construction