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Let n, k ∈ ℕ with 2 kn and X be an n -set. The Enomoto-Katona space

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{R}: = \left\{ {\{ A,B\} \subseteqq \left( {_k^X } \right)\left| {A \cap } \right.B = \not 0} \right\},$$ \end{document}
consisting of all unordered pairs of disjoint k -element subsets of X and equipped with
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$d^\mathcal{R}$$ \end{document}
({ A,B }, { S,T }) ≔ min {| A \ S |+| B \ T |,| A \ T |+| B \ S |}, is considered. The proof of the triangle inequality for
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$d^\mathcal{R}$$ \end{document}
is simplified. Upper bounds on the coding type problem, i.e. the determination of the maximum cardinality of a code consisting of unordered pairs of subsets far away from each other, are improved. The sphere packing problem, i.e. the determination of the maximum number of disjoint balls of a prescribed radius, is introduced and discussed. It is less closely connected to the first problem than it is in the most important spaces of coding theory.

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