P. Frankl. On the trace of finite sets. Journal of Combinatorial Theory(A), 34:41--45, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....friends, then the conditions of the Friendship Theorem fail to hold. Because this proof only uses direct reasoning, and a counting argument, it can be formalized with polynomial size Frege proofs. 3. 2 FrankFs Theorem Another potential hard example is the propositional version of Frankl s theorem [14] stated next. 2 ) Then for Theorem 10 Let t be a positive integer and let m n t any m x n matriz of distinct rows of 0 s and 1 , there is a column such that, if this column is deleted, the resulting m (n 1) matriz will contain at most 2 t 1 1 pairs of equal rows. The tautologies ....

....at most 2 t 1 1 pairs of equal rows. The tautologies based on Frankl s theorem do have polynomial size extended Frege proofs; however, it is an open question whether they have polynomial or quasipolynomial size Frege 1 oofs. The only proof of Frankl s Theorem that we know of is due to Frankl [14], and a brief outline of his proof can be given as follows. Define a 0 1 matrix to be hereditary if all its rows are distinct and changing any 1 entry to a 0 causes two rows to become identical. Frankl first argues that it suffices to prove Theorem 10 for hereditary matrices by proving that any ....

[Article contains additional citation context not shown here]

P. FRANKL, On the trace of finite sets, J. Combin. Theory A, 34 (1983), pp. 41 45.

....matrix of ones forb(m, F) m k m k 1 . m 0 =#(m k ) Also if F has no repeated columns, then forb(m, F ) # m k 1 m k 2 . m 0 =#(m k 1 ) but the bound need not be best possible. For other results or generalizations see [1, 2, 6, 7]. 2 Linear Bounds Careful structural analysis has resulted in bounds that are new and surprisingly exact. The following graph theory argument is the key to the exact bound in the Theorem 2.2 that follows. One can envision this as a generalization of Redei s result [11] that a tournament has a ....

P. Frankl, On the trace of finite sets, J. Combin. Th. (A) 34, (1983), 41-45.

.... m k m k Gamma 1 : m 0 = Theta(m k ) Also if F has no repeated columns, then forb(m; F ) m k Gamma 1 m k Gamma 2 : m 0 = Theta(m k Gamma1 ) but the bound need not be best possible. For other results or generalizations see [1, 2, 6, 7]. 2 Linear Bounds Careful structural analysis has resulted in bounds that are new and surprisingly exact. The following graph theory argument is the key to the exact bound in the Theorem 2.2 that follows. One can envision this as a generalization of R edei s result [11] that a tournament has a ....

P. Frankl, On the trace of finite sets, J. Combin. Th. (A) 34, (1983), 41-45.

....not all in the same language) To date no bound better than the rather trivial 1=n has been obtained by what could be considered a purely combinatorial argument. We should also mention here that for very small p (up to 2 Gamman=2 ) the best possible results on the max norm are available. Frankl [Fr] solved the problem using the Kruskal Katona Theorem. We do not see how to extend this to larger p: For the more restricted class of f 0 s corresponding to intersecting families of subsets (in the language of game theory, symmetric games) an equivalent version of the problem of minimizing the ....

Frankl, P.: On the trace of finite sets. Jour. Comb. Th. ser. A 34(1983) pp. 41-45.

....lemma (as it has become known) states that any set system A traces at least jAj sets in P(n) This formulation is due to Pajor [12] In particular, if jAj P k i=0 Gamma n i Delta , then A must trace some set of size k. This theorem leads naturally in several directions. Both Frankl [9] and Dudley [7] have characterised maximal systems which cover no 2 set, and several authors have considered defect Sauer results. Such results address the problem of determining how large a set system A ae P(n) must be before one is guaranteed a trace of at least M on some k set. We write tr ....

....[4] we present a new criterion for a system to be extremal and analyse the relationship between Sauer s lemma and the Reverse Kleitman Inequality of [4] In the later sections we consider various cases of the defect Sauer problem. First we prove an extension of some of the results of Frankl s [9] and in the final section we turn to the case where N is a polynomial function of n and k is proportional to n. In the positive direction we show that if r is fixed, ff 2 (0; 1) and n 1 then (n r ; n) Gamma (1 Gamma o(1) n r ; ffn Delta : 2 Here is a function of ff only. We also ....

[Article contains additional citation context not shown here]

P. Frankl, On the trace of finite sets, J. Comb. Theory Ser. A 34 (1983), 41--45.

....this time we do not have a good guess as to what the best possible constant is. This remains an intriguing open problem. It is also open 2 whether similar results hold for any of the various generalizations of the Vapnik Chervonenkis dimension and the Sauer VC lemma that have been studied (e.g. [Fra83,Dud87,Pol90,Hau91,HL91]) 2 Proofs of the Results Throughout this section we assume that V f0; 1g n and the Vapnik Chervonenkis dimension of V is d. We begin with the following simple lemma from [HLW90] Let E be the set of all pairs ( u; v) with u; v 2 V such that ae( u; v) 1=n. Thus E is the set of edges in ....

Peter Frankl. On the trace of finite sets. Journal of Combinatorial Theory(A), 34:41--45, 1983.

.... testing [RV] In combinatorics of hypergraphs, set systems of VC dimension d can be viewed as a class of hypergraphs with a certain forbidden subhypergraph (the complete hypergraph on d 1 points) which puts this topic into a broader context of extremal hypergraph theory (see for instance [Fra83] WF94] DSW94] Here we do not consider these areas. This survey is mainly focused on the directions of author s own work; we review some general results of combinatorial nature about set systems of bounded VC dimension, and present applications in geometric discrepancy theory, in ....

P. Frankl. On the trace of finite sets. J. Comb. Theory, Ser. A, 34:41--45, 1983.

....friends, then the conditions of the Friendship Theorem fail to hold. Because this proof only uses direct reasoning, and a counting argument, it can be formalized with polynomial size Frege proofs. 3. 2 Frankl s Theorem Another potential hard example is the propositional version of Frankl s theorem [14] stated next. Theorem 10 Let t be a positive integer and let m n (2 t Gamma1) t . Then for any m Theta n matrix of distinct rows of 0 s and 1 s, there is a column such that, if this column is deleted, the resulting m Theta (n Gamma 1) matrix will contain at most 2 t Gamma1 Gamma 1 ....

....2 t Gamma1 Gamma 1 pairs of equal rows. The tautologies based on Frankl s theorem do have polynomial size extended Frege proofs; however, it is an open question whether they have polynomial or quasipolynomial size Frege proofs. The only proof of Frankl s Theorem that we know of is due to Frankl [14], and a brief outline of his proof can be given as follows. Define a 0=1 matrix to be hereditary if all its rows are distinct and changing any 1 entry to a 0 causes two rows to become identical. Frankl first argues that it suffices to prove Theorem 10 for hereditary matrices by proving that any ....

[Article contains additional citation context not shown here]

P. Frankl, On the trace of finite sets, J. Combin. Theory A, 34 (1983), pp. 41--45.

....= fv[x 1 ; x k ] v 2 V g: ffl g V (k) denotes the maximum cardinality of any projection of V onto k coordinates; i.e. g V (k) maxfjP I (V )j : I f1; ng jIj = kg: Note that g V (n) jV j. By convention, g V (0) 1 and g V (m) jV j, for m n. Frankl s arrow relation [20] is very useful when we know g V (s) and want to bound jV j. Notation 2.2 Let m; n; r; s be positive integers with n s. m; n) r; s) 8V f0; 1g n ) jV j m ) g V (s) r] To restate, m; n) r; s) means that for every V f0; 1g n if g V (s) r Gamma 1 then jV j m Gamma 1. ....

....on f0; 1g n , i.e. a 1 ; a n ) b 1 ; b n ) iff a i b i for i = 1; n. ffl We say that V is closed iff V is closed downwards under , i.e. iff (v w w 2 V ) v 2 V . Note that 0 n belongs to every closed closed subset of f0; 1g n . As pointed out in [1, 20], Fact 2.4 is a corollary of the following important result which was discovered independently by Alon [1] and Frankl [20] Fact 2.6 For every V f0; 1g n there is a closed set W f0; 1g n such that jW j = jV j and g W (i) g V (i) for all i 0: The point is that Fact 2.6 greatly ....

[Article contains additional citation context not shown here]

P. Frankl. On the trace of finite sets. J. of Combinatorial Theory, Ser. A, 34, 41--45, 1983.

No context found.

P. Frankl. On the trace of finite sets. Journal of Combinatorial Theory(A), 34:41--45, 1983.

No context found.

P. Frankl. On the trace of finite sets. Journal of Combinatorial Theory(A), 34:41--45, 1983.

No context found.

Frankl, P. (1983) On the trace of finite sets. Journal of Combinatorial Theory(A), 34, 41.

No context found.

P. Frankl. On the trace of finite sets. J. Combin. Theory, 34:41--45, 1983.

No context found.

P. Frankl, On the trace of finite sets, J. Combin. Theory A, 34 (1983), pp. 41--45.

No context found.

P. Frankl, On the trace of finite sets, J. Combin. Th. (A) 34, (1983), 41-45.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC