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## ASYMPTOTIC UPPER BOUNDS ON THE SHADES OF t-INTERSECTING FAMILIES (2008)

Citations: | 2 - 1 self |

### Citations

325 | Intersection theorems for systems of finite sets
- Erdős, Ko, et al.
- 1961
(Show Context)
Citation Context ...efine the function (6) M(n, k, t) = max A∈I(n,k,t) |A|. The investigation into the function M and the structure of the maximal families was initiated by Erdős–Ko–Rado in 1938, but not published until =-=[EKR61]-=-. In this paper, they gave a complete solution for the case t = 1, and posed what became one of the most famous open problems in this area. The following so called 4mconjecture for the case t = 2: (7)... |

137 |
Combinatorics of Finite Sets.
- Anderson
- 1987
(Show Context)
Citation Context ...nection is described in lemma 1 below (cf. [Hir08] for details). 1.1. Shades. One of the basic notions in Sperner theory is the shade (also called upper shadow) of a set or a family of sets (see e.g. =-=[And02]-=-,[Eng97]). For a subset x of a fixed set S, the shade of x is (1) ∇(x) = {y ⊆ S : x ⊂ y and |y| = |x| + 1}, and the shade of a family X of subsets of S is (2) ∇(X) = ⋃ ∇(x). x∈X Date: August 10, 2008.... |

103 | The complete intersection theorem for systems of finite sets
- Ahlswede, Khachatrian
- 1997
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Citation Context ...s generalized by Frankl in 1978 ([Fra78]) as follows: For all 1 ≤ t ≤ k ≤ n, (10) M(n, k, t) = max |Fi(n, k, t)|. 0≤i≤ n−t 2 In 1995, the general conjecture was proven true by Ahlswede–Khachatrian in =-=[AK97]-=-, where they moreover established that the optimal families in I(n, k, t) are equal to one of the families Fi(n, k, t) up to a permutation of [n]. This finally settled the 4m-conjecture, and moreover ... |

44 |
The Erdős-Ko-Rado Theorem is true for n = ckt,
- Frankl
- 1978
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Citation Context ... in m := 2m (i.e. 2m for m), s := 1 and i := m − 1 we see that the right hand side of equation (7) is equal to the cardinality of Fm−1(4m, 2m, 2). The 4m-conjecture was generalized by Frankl in 1978 (=-=[Fra78]-=-) as follows: For all 1 ≤ t ≤ k ≤ n, (10) M(n, k, t) = max |Fi(n, k, t)|. 0≤i≤ n−t 2 In 1995, the general conjecture was proven true by Ahlswede–Khachatrian in [AK97], where they moreover established ... |

27 |
Sperner Theory. Encyclopedia of Mathematics and Its Applications 65.
- Engel
- 1997
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Citation Context ...is described in lemma 1 below (cf. [Hir08] for details). 1.1. Shades. One of the basic notions in Sperner theory is the shade (also called upper shadow) of a set or a family of sets (see e.g. [And02],=-=[Eng97]-=-). For a subset x of a fixed set S, the shade of x is (1) ∇(x) = {y ⊆ S : x ⊂ y and |y| = |x| + 1}, and the shade of a family X of subsets of S is (2) ∇(X) = ⋃ ∇(x). x∈X Date: August 10, 2008. 2000 Ma... |

16 | special are Cohen and random forcings, i.e. Boolean algebras of the family of subsets of reals modulo meagre or null
- Shelah, How
- 1994
(Show Context)
Citation Context ...f set theoretic forcing, of whether Cohen and random forcing together form a basis for all nontrivial Souslin (i.e. “simply” definable) ccc (i.e. no uncountable antichains) posets (asked by Shelah in =-=[She94]-=-). Dichotomy 1. Every analytic (i.e. projection of a closed subset of the “plane”) family A of infinitely branching subtrees of {0, 1} <N satisfies at least one of the following: (a) There exists a co... |

5 |
and Norihide Tokushige. The exact bound in the Erdős-Ko-Rado theorem for cross-intersecting families
- Matsumoto
- 1989
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Citation Context ...e are a number of results on cross-t-intersecting families in the literature; however, the state of knowledge seems very meager compared with t-intersecting families. The following theorem, proved in =-=[MT89]-=-, is the strongest result of its kind that we were able to find. Theorem 2 (Matsumoto–Tokushige, 1989). N(n, k, l, 1) = ( )( ) n−1 n−1 k−1 l−1 whenever 2k, 2l ≤ n. Note that this corresponds to case t... |

1 | Nonhomogeneous analytic families of trees
- Hirschorn
- 2008
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Citation Context ...bounds on the cardinalities of m-shades of t(m)-intersecting families of k(m)subsets of [2m], as m → ∞. A generalization to cross-t-intersecting families is also considered. 1. Introduction The paper =-=[Hir08]-=- was concerned with the dichotomy below of descriptive set theory. This dichotomy is aimed towards research on a fundamental question of set theoretic forcing, of whether Cohen and random forcing toge... |

1 |
An upper bound on the capacity of the boundary of an antichain in an n-dimensional cube, Diskret
- Kostochka
- 1989
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Citation Context ...there are numerous results giving lower bounds on the size of shadows/shades, upper bounds seem to be rather scarce. Perhaps this is because they are not very good. For example, the following is from =-=[Kos89]-=-, where 2 S denotes the power set of S. Theorem 1 (Kostochka, 1989). Suppose that A ⊆ 2 [n] is a Sperner family. Then ∇(A) ≤ 0.724 · 2 n . Moreover, the best upper bound is known to be greater than 0.... |

1 |
Introduction to mathematical probability., IX+ 411 p
- Uspensky
- 1937
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Citation Context ...orth, we shall make the simplification that k(m) = o(m), i.e. limm→∞ k(m) / m = 0; this was the only case used in our application. Before proceeding further, recall the de Moivre–Laplace theorem (cf. =-=[Usp37]-=-), roughly stating that the binomial series of (p+q) n has most of the sum concentrated in the order of √ n terms around the center: For all 0 ≤ a, b < ∞, (15) lim where ⌊b √ n/2⌋ ∑ n→∞ j=−⌊a √ n/2⌋ (... |