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A block design is a pair (V,B), such that V is a finite non-empty set and B is a finite collection of non-empty subsets of V, called blocks, such that every element of V is in at least one block. In the collection of blocks, the order does not matter, but the number of times a block occurs (its multiplicity) does matter. In technical terms, the collection of blocks is a multiset. An important and interesting class of block designs is that of t-designs (see my project on t-designs). For t a non-negative integer, a t-design, or more specifically a t-(v,k,λ) design, is a block design (V,B), such that V has exactly v elements, each block has the same size k, and each t-element subset of V is contained in the same positive number λ of blocks. Now, given positive integers t, v, k, λ, with t>1, it can be very difficult (often an open research problem) to determine whether or not a t-(v, k, λ) design exists. In addition, when such a design exists, it may or may not have repeated blocks. There has been some recent work on the following problem. Given positive integers t, v, k, λ, with t>1, determine an upper bound on the multiplicity of any block in any t-(v, k, λ) design. It is interesting to know if and when such a bound is met for some block in some t-(v, k, λ) design.
This project will be based on the study of some recent papers [1,2,3]. One interesting task would be to determine, for a list of possible parameter tuples (t, v, k, λ) (for reasonably sized parameter values), what can be said about the maximum multiplicity of a block in any t-(v, k, λ) design.
- P. J. Cameron, L. H. Soicher, Block intersection polynomials, Bull. London Math. Soc. 39 (2007) 559–564.
- P. Dobcsányi, D. A. Preece, L. H. Soicher, On balanced incomplete-block designs with repeated blocks, European J. Combinatorics 28 (2007) 1955–1970.
- J. P. McSorley, L. H. Soicher, Constructing t-designs from t-wise balanced designs, European J. Combinatorics 28 (2007) 567–571.
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