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The complete nontrivial–intersection theorem for systems of finite sets
 J. Combin.Theory Ser.A
, 1996
"... The authors have proved in a recent paper a complete intersection theorem for systems of finite sets. Now we establish such a result for nontrivialintersection systems (in the sense of Hilton and Milner [Quart. J. Math. Oxford 18 (1967), 369 384]. 1996 Academic Press, Inc. 1. ..."
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The authors have proved in a recent paper a complete intersection theorem for systems of finite sets. Now we establish such a result for nontrivialintersection systems (in the sense of Hilton and Milner [Quart. J. Math. Oxford 18 (1967), 369 384]. 1996 Academic Press, Inc. 1.
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Cited by 11 (0 self)
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
A HiltonMilner Theorem for Vector Spaces
"... We show for k � 2 that if q � 3 and n � 2k + 1, or q = 2 and n � 2k + 2, then any intersecting family F of ksubspaces of an ndimensional vector space over GF(q) with ⋂ F ∈F F = 0 has size at most [] n−1 k−1 − qk(k−1) [] n−k−1 k−1 + qk. This bound is sharp as is shown by HiltonMilner type familie ..."
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We show for k � 2 that if q � 3 and n � 2k + 1, or q = 2 and n � 2k + 2, then any intersecting family F of ksubspaces of an ndimensional vector space over GF(q) with ⋂ F ∈F F = 0 has size at most [] n−1 k−1 − qk(k−1) [] n−k−1 k−1 + qk. This bound is sharp as is shown by HiltonMilner type families. As an application of this result, we determine the chromatic number of the corresponding qKneser graphs.
Forbidden (0,1)vectors in hyperplanes of R n : the restricted case
 Designs, Codes and Cryptogr
, 2003
"... Abstract. In this paper we continue our investigation on “Extremal problems under dimension constraint ” introduced in [2]. Let E(n, k) be the set of (0,1)vectors in R n with k one’s. Given 1 ≤ m, w ≤ n let X ⊂ E(n, m) satisfy span(X) ∩ E(n, w) = ∅. Howbig can X  be? This is the main problem st ..."
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Abstract. In this paper we continue our investigation on “Extremal problems under dimension constraint ” introduced in [2]. Let E(n, k) be the set of (0,1)vectors in R n with k one’s. Given 1 ≤ m, w ≤ n let X ⊂ E(n, m) satisfy span(X) ∩ E(n, w) = ∅. Howbig can X  be? This is the main problem studied in this paper. We solve this problem for all parameters 1 ≤ m, w ≤ n and n> n0(m, w).
BRACE–DAYKIN TYPE INEQUALIES FOR INTERSECTING FAMILIES
"... ABSTRACT. Let n,k and r ≥ 8 be positive integers. Suppose that a family F ⊂ � [n] � k satisfies F1 ∩···∩Fr � = /0 for all F1,...,Fr ∈ F and � F∈F F = /0. We prove that there exist εr> 0 and nr such that n − r − 1 n − r − 1 F  ≤ (r + 1) k − r k − r − 1 holds for all n and k, satisfying n> n ..."
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ABSTRACT. Let n,k and r ≥ 8 be positive integers. Suppose that a family F ⊂ � [n] � k satisfies F1 ∩···∩Fr � = /0 for all F1,...,Fr ∈ F and � F∈F F = /0. We prove that there exist εr> 0 and nr such that n − r − 1 n − r − 1 F  ≤ (r + 1) k − r k − r − 1 holds for all n and k, satisfying n> nr and  k n − 1 2  < εr. 1.
Documenta Math. 65 On the Number of Square Classes of a Field of Finite Level
, 2001
"... Abstract. The level question is, whether there exists a field F with finite square class number q(F): = F × /F ×2  and finite level s(F) greater than four. While an answer to this question is still not known, one may ask for lower bounds for q(F) when the level is given. For a nonreal field F of l ..."
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Abstract. The level question is, whether there exists a field F with finite square class number q(F): = F × /F ×2  and finite level s(F) greater than four. While an answer to this question is still not known, one may ask for lower bounds for q(F) when the level is given. For a nonreal field F of level s(F) = 2 n, we consider the filtration of the groups DF (2 i), 0 ≤ i ≤ n, consisting of all the nonzero sums of 2 i squares in F. Developing further ideas of A. Pfister, P. L. Chang and D. Z. Djoković and by the use of combinatorics, we obtain lower bounds for the invariants q i: = DF (2 i)/DF (2 i−1), for 1 ≤ i ≤ n, in terms of s(F). As a consequence, a field with finite level ≥ 8 will have at least 512 square classes. Further we give lower bounds on the cardinalities of the Witt ring and of the 2torsion part of the Brauer group of such a field. 1
A new short proof of a theorem of Ahlswede and Khachatrian
, 2007
"... Ahlswede and Khachatrian [5] proved the following theorem, which answered a question of Frankl and Füredi [3]. Let 2 ≤ t + 1 ≤ k ≤ 2t + 1 and n ≥ (t + 1)(k − t + 1). Suppose that F is a family of ksubsets of an nset, every two of which have at least t common elements. If  ∩F ∈F F  < t, then  ..."
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Ahlswede and Khachatrian [5] proved the following theorem, which answered a question of Frankl and Füredi [3]. Let 2 ≤ t + 1 ≤ k ≤ 2t + 1 and n ≥ (t + 1)(k − t + 1). Suppose that F is a family of ksubsets of an nset, every two of which have at least t common elements. If  ∩F ∈F F  < t, then F  ≤ (t + 2) � � � � n−t−2 n−t−2 k−t−1 + k−t−2, and this is best possible. We give a new, short proof of this result. The proof in [5] requires the entire machinery of the proof of the complete intersection theorem, while our proof uses only ordinary compression and an earlier result of Wilson [7]. 1