## Probabilistic Methods in Extremal Finite Set Theory (1991)

Venue: | Miklós Eds.), Bolyai Society Mathematical Studies,3, Visegrád |

Citations: | 12 - 2 self |

### BibTeX

@INPROCEEDINGS{Alon91probabilisticmethods,

author = {Noga Alon},

title = {Probabilistic Methods in Extremal Finite Set Theory},

booktitle = {Miklós Eds.), Bolyai Society Mathematical Studies,3, Visegrád},

year = {1991},

pages = {39--57}

}

### OpenURL

### Abstract

There are many known applications of the Probabilistic Method in Extremal Finite Set Theory. In this paper we describe several examples, demonstrating some of the techniques used and illustrating some of the typical results obtained. This is partly a survey paper, but it also contains various new results.

### Citations

1934 | Random Graphs
- Bollobás
- 1985
(Show Context)
Citation Context ...e Pr(A ∈ F) ≤ k/n. But A is uniformly chosen from all k-sets so implying that ✷ k n |F| ≥ Pr(A ∈ F) = �n� , |F| ≤ k n � � n = k k � � n − 1 . k − 1 We conclude this section with a theorem of Bollobás =-=[6]-=-. The proof presented here is due to Jaeger and Payan [17] and Katona [19]. Let F = {(Ai, Bi)} h i=1 be a family of pairs of subsets of an arbitrary set. We call F a (k, l)-system if |Ai| = k and |Bi|... |

769 |
A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann
- Chernoff
- 1952
(Show Context)
Citation Context ... i ≤ n, the result follows from Proposition 4.1. ✷ 13sAs observed by Peter Frankl [15], the last corollary supplies a quick proof for the well known estimate, (that follows, e.g., from the results in =-=[9]-=-), that for every integer n and for every real 0 < p ≤ 0.5, � � i≤np ≤ 2nH(p) . Indeed, let F be the family of all subsets of cardinality at most � n i pn of {1, 2, . . . , n}. If pi is the fraction o... |

203 |
intersection theorems for systems of finite sets
- Erdős, Ko, et al.
- 1961
(Show Context)
Citation Context ...e an antichain. Then |F| ≤ � n � ⌊n/2⌋ Proof The function �n� x is maximized at x = ⌊n/2⌋ so that ✷ 1 ≥ � A∈F 1 � n |A| � ≥ |F| � n The second basic result presented here is the Erdős Ko Rado Theorem =-=[13]-=-. A family F of sets is called intersecting if for every A, B ∈ F A ∩ B �= ∅. 2 ⌊n/2⌋ �.sTheorem 1.3 ([13]) Suppose n ≥ 2k and let F be an intersecting family of k-element subsets of � . an n-set. The... |

140 | A lower bound for radio broadcast
- Alon, Bar-Noy, et al.
- 1991
(Show Context)
Citation Context ...t hits H. Finally, let t(n) denote the maximum possible value of t(H), where the maximum is taken over all families H of n subsets of n. Our objective is to estimate t(n). This problem, considered in =-=[1]-=-, is motivated by the study of a certain communication network, as we briefly describe below. A radio network is a synchronous network of processors that communicate by transmitting messages to their ... |

121 | A graph-theoretic game and its application to the k-server problem
- Alon, Karp, et al.
- 1995
(Show Context)
Citation Context ...|F|, whereas H(X(Gi)) ≤ log 2 |Fi|, implying the desired result. ✷ 14sA special case of the last corollary is proved in [23], in a different method. Another application of Proposition 4.3 is given in =-=[4]-=-. The d-dimensional grid Gn,d is the graph formed by the product of d n-vertex paths. It has N = n d vertices and dn d−1 (n − 1) edges. For a spanning tree T of G = Gn,d, let V (T ) denote the average... |

56 | Some intersection theorems for ordered sets and graphs
- Chung, Graham, et al.
- 1986
(Show Context)
Citation Context ...H(p) is increasing for 0 ≤ p ≤ 0.5 this, together with Corollary 4.2 implies that � � � n �n = |F| ≤ 2 i=1 i H(pi) nH(p) ≤ 2 , as needed. i≤np An interesting extension of Proposition 4.1 is proved in =-=[10]-=-. As in that proposition, let X = (X1, . . . , Xn) be a random variable taking values in the set S = S1 × S2 × . . . × Sn, where each Xi is a random variable taking values in Si. For a subset I of {1,... |

47 |
An inequality related to the isoperimetric inequality
- Loomis, Whitney
- 1949
(Show Context)
Citation Context ...probability 1/|F|. By Proposition 4.3 m� kH(X) ≤ H(X(Gi)). i=1 But H(X) = log 2 |F|, whereas H(X(Gi)) ≤ log 2 |Fi|, implying the desired result. ✷ 14sA special case of the last corollary is proved in =-=[23]-=-, in a different method. Another application of Proposition 4.3 is given in [4]. The d-dimensional grid Gn,d is the graph formed by the product of d n-vertex paths. It has N = n d vertices and dn d−1 ... |

44 |
Families of k-independent sets
- Kleitman, Spencer
- 1973
(Show Context)
Citation Context ... n} is k-independent if for every k distinct sets F1, F2, . . . , Fk of F, all the 2 k intersections ∩ k i=1 Gi are nonempty, where each Gi is either Fi or its complement N \ Fi. Kleitman and Spencer =-=[21]-=- considered the problem of estimating the maximum possible cardinality of a k-independent family of subsets of an n-set. Their lower bound is proved by a random construction. Theorem 2.1 If � � m 2 k ... |

28 |
Explicit construction of exponential sized families of k-independent sets
- Alon
- 1986
(Show Context)
Citation Context ... known explicit construction of such a large k-independent family, although there are known explicit constructions of such families of size 2 ckn , where ck > 0 is a constant depending only on k. See =-=[2]-=- for more details. The second random construction we describe, due to Erdős and Fűredi [12], is similar, but has an interesting application in Combinatorial Geometry. Proposition 2.2 For every n ≥ 1 t... |

27 |
Über zwei Probleme bezüglich konvexer Körper von P. Erdös und von
- Danzer, Grünbaum
- 1962
(Show Context)
Citation Context ...nts in the n-dimensional Euclidean space R n , such that all angles determined by three points from the set are strictly less than π/2. This theorem disproves an old conjecture of Danzer and Grűnbaum =-=[11]-=-, that the maximum cardinality of such a set is at most 2n − 1. We note that as proved by Danzer and Grűnbaum the maximum cardinality of a set of points in R n in which all angles are at most π/2 is 2... |

19 |
Private communication
- Fiat
(Show Context)
Citation Context ...cond example we describe in this section is a very recent result of the author, and it is very likely that the estimate here can be still improved. The problem we consider was raised by Fiat and Naor =-=[14]-=-, who were motivated by the study of a method for distributing keys in a certain multi-user crypto-system. The objective is, roughly, to distribute keys among n users, so that each one receives a smal... |

13 | The greatest angle among n points in the d-dimensional Euclidean space
- Erdős, Füredi
- 1983
(Show Context)
Citation Context ...n explicit constructions of such families of size 2 ckn , where ck > 0 is a constant depending only on k. See [2] for more details. The second random construction we describe, due to Erdős and Fűredi =-=[12]-=-, is similar, but has an interesting application in Combinatorial Geometry. Proposition 2.2 For every n ≥ 1 there is a family F of m subsets of N = {1, . . . , n}, where m = ⌊ 1 2 2 ( √ ) 3 n⌋, such t... |

4 | The maximum number of disjoint pairs in a family of subsets
- Alon, Frankl
- 1985
(Show Context)
Citation Context ...ős conjectured that if F is a family of m subsets of X, and m = 2 (1/(k+1)+δ)n , where δ > 0, then d(F) ≤ (1 − 1 k ) � � m 2 as n tends to infinity. + o(m 2 ) The more general conjecture is proved in =-=[3]-=-. Since the proof for the general case is somewhat complicated, we describe here only that of the special case k = 1, mentioned above. We prove a stronger result, as follows. 1 ( Theorem 3.1 ( [3]) Le... |

3 |
Solution of a problem of Ehrenfeucht
- KATONA
- 1974
(Show Context)
Citation Context ...that ✷ k n |F| ≥ Pr(A ∈ F) = �n� , |F| ≤ k n � � n = k k � � n − 1 . k − 1 We conclude this section with a theorem of Bollobás [6]. The proof presented here is due to Jaeger and Payan [17] and Katona =-=[19]-=-. Let F = {(Ai, Bi)} h i=1 be a family of pairs of subsets of an arbitrary set. We call F a (k, l)-system if |Ai| = k and |Bi| = l for all 1 ≤ i ≤ k, Ai ∩ Bi = ∅ and Ai ∩ Bj �= ∅ for all 1 ≤ i, j ≤ k.... |

2 |
Private communication
- Frankl
- 1994
(Show Context)
Citation Context ...F )) = 1/|F| for all F ∈ F. Clearly H(X) = |F|(− 1 1 |F| log |F| ) = log |F|, and since here H(Xi) = H(pi) for all 1 ≤ i ≤ n, the result follows from Proposition 4.1. ✷ 13sAs observed by Peter Frankl =-=[15]-=-, the last corollary supplies a quick proof for the well known estimate, (that follows, e.g., from the results in [9]), that for every integer n and for every real 0 < p ≤ 0.5, � � i≤np ≤ 2nH(p) . Ind... |

2 |
Intersection of k-element sets, Combinatorica 1
- Kleitman, Shearer, et al.
- 1981
(Show Context)
Citation Context ...x1, . . . , xn) p(1 : x1)p(2 : x2) . . . p(n : xn)( ) = 1. p(1 : x1)p(2 : x2) . . . p(n : xn) Therefore H(X) − � n i=1 H(Xi) ≤ −1 log 2 1 = 0, completing the proof. ✷ The above proposition is used in =-=[20]-=- to derive several interesting applications in Extremal Finite Set Theory, including an upper estimate for the maximum possible cardinality of a family of k-sets in which the intersection of no two is... |

1 |
An average distance inequality for subsets of the cube
- Althőfer, Sillke
(Show Context)
Citation Context ...ance in A, denoted dist(A), dist(A) = 1 |A| 2 � A∈A B∈A � d(A, B), where d(A, B) is the cardinality of the symmetric difference of A and B (i.e., the cardinality of (A \ B) ∪ (B \ A)). The authors of =-=[5]-=- proved that for every family A as above, dist(A) ≥ n + 1 2 − 2n−1 |A| . This result is sharp for |A| = 2 n−1 (and, of course, for |A| = 2 n ), but it is very far from being sharp for smaller values o... |

1 |
Broadcast in radio networks; an exponential gap between determinism and randomization
- Bar-Yehuda, Goldreich, et al.
- 1986
(Show Context)
Citation Context ...es to their neighbors. A processor receives a message in a given step iff it is silent in this step and precisely one of its neighbors transmits. This model is discussed in various papers, see, e.g., =-=[8]-=-, [1] and their references. Suppose that a processor p in this model 6shas a message which it has to broadcast to all the other processors in the network. Suppose also, for simplicity, that p has a co... |

1 |
Nombre maximal d’arétes d’un hypergrphe critique de rang h
- Jaeger, Payan
- 1971
(Show Context)
Citation Context ...ets so implying that ✷ k n |F| ≥ Pr(A ∈ F) = �n� , |F| ≤ k n � � n = k k � � n − 1 . k − 1 We conclude this section with a theorem of Bollobás [6]. The proof presented here is due to Jaeger and Payan =-=[17]-=- and Katona [19]. Let F = {(Ai, Bi)} h i=1 be a family of pairs of subsets of an arbitrary set. We call F a (k, l)-system if |Ai| = k and |Bi| = l for all 1 ≤ i ≤ k, Ai ∩ Bi = ∅ and Ai ∩ Bj �= ∅ for a... |

1 | A simple proof of the Erdős Ko Rado Theorem - Katona - 1972 |

1 |
Parallel two-prover zero-knowledge protocols
- Lapidot, Shamir
(Show Context)
Citation Context ... its proof are similar to the ones described in this section. This result answers a question of Lapidot and Shamir, and although it may look somewhat artificial it is naturally suggested, as shown in =-=[22]-=-, in the study of the possibility to parallelize certain two-prover zero-knowledge protocols. 16sLet Z n k ×Zn k denote the set of all ordered pairs (u, v), where u and v are vectors of length n over ... |

1 |
Ein Satz über Untermengeneiner endlichen
- Sperner
- 1928
(Show Context)
Citation Context ...has numerous generalizations, extensions and applications. Their simple probabilistic proofs demonstrate nicely the basic application of probabilistic arguments. The first result is Sperner’s Theorem =-=[25]-=-, considered by many researchers to be the starting point of Extremal Finite Set Theory. Recall that a family F of subsets of {1, . . . , n} is called an antichain if no set of F is contained in anoth... |