Abstract
Let X be an n-element finite set, 0 < k ≤ n/2 an integer. Suppose that {A 1, A 2} and {B 1, B 2} are pairs of disjoint k-element subsets of X (that is, ∣A 1∣ = ∣A 2∣ = ∣B 1∣ = B 2∣ = k, A 1 ∩A 2 = ∅, B 1 ∩ B 2 = ∅). Define the distance of these pairs by d({A 1 ∣, A 2},{B 1, B 2}) = min {∣A 1 − B 1 ∣ + ∣A 2 − B 2∣, ∣A 1 − B 2∣+∣A 2 − B 1∣}. This is the minimum number of elements of A 1 ∪ A 2 one has to move to obtain the other pair {B 1, B 2}. Let C(n, k, d) be the maximum size of a family of pairs of disjoint k-subsets, such that the distance of any two pairs is at least d.
Here we establish a conjecture of Brightwell and Katona concerning an asymptotic formula for C(n,k, d) for k, d are fixed and n → ∞. Also, we find the exact value of C(n, k, d) in an infinite number of cases, by using special difference sets of integers. Finally, the questions discussed above are put into a more general context and a number of coding theory type problems are proposed.
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Bollobás, B., Füredi, Z., Kantor, I., Katona, G.O.H., Leader, I. (2014). A coding problem for pairs of subsets. In: Matoušek, J., Nešetřil, J., Pellegrini, M. (eds) Geometry, Structure and Randomness in Combinatorics. CRM Series, vol 18. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-525-7_4
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DOI: https://doi.org/10.1007/978-88-7642-525-7_4
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