# Two-Part Set Systems

@article{Erds2012TwoPartSS,
title={Two-Part Set Systems},
author={P{\'e}ter L. Erd{\"o}s and D{\'a}niel Gerbner and Nathan Lemons and Dhruv Mubayi and Cory Palmer and Bal{\'a}zs Patk{\'o}s},
journal={Electron. J. Comb.},
year={2012},
volume={19},
pages={P52}
}
The two part Sperner theorem of Katona and Kleitman states that if $X$ is an $n$-element set with partition $X_1 \cup X_2$, and $\mathcal{F}$ is a family of subsets of $X$ such that  no two sets $A, B \in \mathcal{F}$  satisfy $A \subset B$ (or $B \subset A$) and $A \cap X_i=B\cap X_i$ for some $i$, then $|\mathcal{F}| \le {n \choose \lfloor n/2\rfloor}$. We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfy… Expand
1 Citations
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• 2021
In this article we investigate a problem in graph theory, which has an equivalent reformulation in extremal set theory similar to the problems researched in [13] by Gyula O.H. Katona, who proposedExpand

#### References

SHOWING 1-10 OF 19 REFERENCES
INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS
• Mathematics
• 1961
2. Notation The letters a, b, c, d, x, y, z denote finite sets of non-negative integers, all other lower-case letters denote non-negative integers. If fc I, then [k, I) denotes the setExpand
On a lemma of Littlewood and Offord
Remark. Choose Xi = l, n even. Then the interval ( — 1, + 1 ) contains Cn,m s u m s ^ i e ^ , which shows that our theorem is best possible. We clearly can assume that all the Xi are not less than 1.Expand
All maximum 2-part Sperner families
• Computer Science, Mathematics
• J. Comb. Theory, Ser. A
• 1986
All families giving equality in this theorem are determined - all families of some subsets of X satisfy the following condition: if there is an inclusion F1 ⊈ F2 (F1, F2 ϵ F ) in F, the difference F2 − F1 cannot be a subset of X1 or X2. Expand
An inequality for the weights of two families of sets, their unions and intersections
• Mathematics
• 1978
then ~(A) fi(B) < 7(A v B) cS(A A B) for all A, B ~ S, (2) where e(A) = ~(a~A) e(a) and A v B = {awb; aeA, b~B} and A A B = {ac~b; a~A, b~B}. Since every distributive lattice can be embedded in theExpand
Extremal problems concerning Kneser graphs
• Computer Science, Mathematics
• J. Comb. Theory, Ser. B
• 1986
It is proven that equality holds for n> 1 2 (3+ 5 )k , and equality holds only if there exist two points a, b such that {a, b} ∩ F ≠ ∅ for all F ∈ A ∪ B. Expand
ON A CONJECTURE OF ERD ˝ OS
• Mathematics
• 2012
Let a be an integer different from 0, 1, or a perfect square. We consider a conjecture of Erd˝ os which states that #f pV'a. p/D rg " r " for any " > 0, where 'a. p/ is the order of a modulo p. InExpand
A tour of M-part L-Sperner families
• Computer Science, Mathematics
• J. Comb. Theory, Ser. A
• 2011
A BLYM inequality for M-part L-Sperner families is proved and results regarding the homogeneity of such families of maximum size are obtained through the convex hull method. Expand
Convex hulls of more-part Sperner families
• Mathematics, Computer Science
• Graphs Comb.
• 1986
The convex hulls of more-part Sperner families is defined and studied and some methods are presented giving new theorems. Expand