# Partition-free families of sets

## Summary (1 min read)

### 1 Introduction

- This famous result served as the starting point of the presently burgeoning field of extremal set theory.
- The authors have a few results concerning this and some related questions that will appear in [10].

### 2 Basic tools

- The following lemma is a generalization of the main lemma from Kleitman’s paper [16].
- Let us choose the pairwise disjoint sets from the claim randomly with uniform distribution.

### 6 The proof of Theorem 4

- For a change, in this section the authors give a proof with a somewhat different and hopefully simpler analysis.
- Exactly three are present (and three are missing).
- Exactly two of the m- and 2m-sets are missing.
- Therefore, the equality cannot hold in this case.
- The authors conclude that they are in the situation c1 and in each triple there in exactly one present m-set and (2m+1)-set, moreover, both belong to the same family.

### 7 Discussion

- One natural direction to extend these results is to study r-partition-free families, defined in the introduction, as well as to study their r-partite analogues.
- Another natural generalization of partition-free families, that was overlooked so far, are the r-box-free families (also defined in the introduction).
- More generally, the authors may ask the following question.
- The following sharp result may be proved using a direct generalization of Kleitman’s argument [16].
- The authors sum up all the obtained inequalities and multiply them by the corresponding ( n s1 ) (except for the first one, which they multiply by 1 2 ( n m ) ).

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##### References

956 citations

### "Partition-free families of sets" refers background in this paper

...E. Sperner, ‘Ein Satz über Untermengen einer endlichen Menge’, Math....

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...K. Engel, Sperner theory, vol. 65 (Cambridge University Press, Cambridge, MA, 1997)....

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...In 1928, Sperner [21] proved that if a family has size greater than ( n n/2 ) , then it must contain two subsets F,G, such that F G....

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...In 1928, Sperner [21] proved that if a family has size greater than ( n n/2 ) , then it must contain two subsets F,G, such that F G....

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519 citations

### "Partition-free families of sets" refers background in this paper

...In connection with an analytic problem of Littlewood and Offord, he proved [4] that if |F| is larger than the sum of the l largest binomial coefficients, then F contains a chain F0 F1 · · · Fl....

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373 citations

### "Partition-free families of sets" refers background in this paper

...For an introduction to the topic, the reader is advised to consult the books [1, 2, 3, 15]....

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285 citations

### "Partition-free families of sets" refers background in this paper

...For an introduction to the topic, the reader is advised to consult the books [1, 2, 3, 15]....

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221 citations

### "Partition-free families of sets" refers background in this paper

...Many of these results and proofs are presented in [13]....

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