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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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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.