Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: A Hilton-Milner Theorem for Vector Spaces. | A Hilton-Milner Theorem for Vector Spaces A. Blokhuis1 A. E. Brouwer1 A. Chowdhury2 P. Frankl3 T. Mussche1 B. Patkós4 and T. Szonyi5 6 1Dept. of Mathematics Technological University Eindhoven . Box 513 5600 MB Eindhoven The Netherlands. 2 Dept. of Mathematics University of California San Diego La Jolla CA 92093 USA. 3ShibuYa-Ku Higashi 1-10-3-301 Tokyo 150 Japan. 4 Department of Computer Science University of Memphis tN 38152-3240 USA. 5 Institute of Mathematics Eotvos Lorand University H-1117 Budapest Pazmany P. s. 1 C Hungary. 6 Computer and Automation Research Institute Hungarian Academy of Sciences H-1111 Budapest Lagymanyosi u. 11 Hungary. aartb@ aeb@ anchowdh@ bpatkos@ tmussche@ szonyi@ Submitted Nov 1 2009 Accepted May 4 2010 Published May 14 2010 Mathematics Subject Classification 05D05 05A30 Abstract 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 k-subspaces of an n-dimensional vector space over CỈ jrtí nHin _ _ ỉr íl l ívK ì í .ì I TTirAcC n 1 _ í k k 1 n k 11 I f fk 11 I Ii RrAnTirl Gr q with rr 0 has size at most R 1 q r. 1 q . This bound I IF Lk 1J L k 1 J is sharp as is shown by Hilton-Milner type families. As an application of this result we determine the chromatic number of the corresponding q-Kneser graphs. 1 Introduction Sets Let X be an n-element set and for 0 k n let X denote the family of all subsets of X of cardinality k. A family F c k is called intersecting if for all F1 F2 E F we have F1 e F2 0. Erdos Ko and Rado 5 determined the maximum size of an intersecting family and introduced the so-called shifting technique. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R71 1 Theorem Erdos-Ko-Rado Suppose F c X is intersecting and n 2k. Then I I in 1A TfvY fri f ybi Y Y r Ti - 0 f 770 Ji IT J iJ 7q Z lTlJlT if sp Ỉ p ÍX p X f I7 1 I. Excepbing the case lb 2k equality Holds only if I F I 7 Ị x __ -L I 1 1 k u
