## Abstract

The two part Sperner theorem of Katona and Kleitman states that if X is an n-element set with partition X_{1} ∪ X_{2}, and F is a family of subsets of X such that no two sets(A, B)∈ F satisfy A ⊂ B (or B ε A) and A ∩ X_{i} = B ∩ X_{i} for some i, then F ≤ n. We consider variations of this problem by replacing the Sperner ⌊n/2⌋ property with the intersection property and considering families that satisfy various combinations of these properties on one or both parts X_{1}, X_{2}. Along the way, we prove the following new result which may be of independent interest: let F, G be intersecting families of subsets of an n-element set that are additionally cross-Sperner, meaning that if A ∈ F and B ∈ G, then A ⊂ B and B ⊂ A. Then |F| + |G| ≤ 2^{n-1} and there are exponentially many examples showing that this bound is tight.

Original language | English |
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Pages (from-to) | 1-10 |

Number of pages | 10 |

Journal | Electronic Journal of Combinatorics |

Volume | 19 |

DOIs | |

State | Published - 2012 |

## Keywords

- Extremal set theory
- Intersecting
- Sperner