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Given a set $X$, a collection $\mathcal{F} \subset \mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this probability.
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Emanuel Sperner · 1928
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Paul Turán · 1941
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P. Erdös · 1945
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P. Erdős, A. Hajnal, and J. W. Moon · 1964
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