An initial segment contains every element below each of its elements.
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- Binary initial segment
- Harper theorem implies the Kruskal-Katona theorem
- Nonunique down-set extremizers for Harper theorem
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- Simplicial order on the discrete cube
- Simplicial section compression
- Space of infinite subsets of the natural numbers
- Terminal families for simplicial section compression
- Vertex-isoperimetric inequality in the discrete cube
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