A Hilton-Milner Theorem for Vector Spaces

Transcription

1 A Hilton-Milner Theorem for ector Spaces A Blohuis, A E Brouwer, A Chowdhury 2, P Franl 3, T Mussche, B Patós 4, and T Szőnyi 5,6 Dept of Mathematics, Technological University Eindhoven, PO Box 53, 5600 MB Eindhoven, The Netherlands 2 Dept of Mathematics, University of California San Diego, La Jolla, CA 92093, USA 3 ShibuYa-Ku, Higashi, Toyo, 50, Japan 4 Department of Computer Science, University of Memphis, TN , USA 5 Institute of Mathematics, Eötvös Loránd University, H-7 Budapest, Pázmány P s /C, Hungary 6 Computer and Automation Research Institute, Hungarian Academy of Sciences, H- Budapest, Lágymányosi ú, Hungary aartb@wintuenl, aeb@cwinl, anchowdh@mathucsdedu, peterfranl@gmailcom, bpatos@memphisedu, tmussche@gmailcom, szonyi@cseltehu Submitted: Nov, 2009; Accepted: May 4, 200; Published: May 4, 200 Mathematics Subject Classification: 05D05, 05A30 Abstract We show for 2 that if q 3 and n 2 +, or q = 2 and n 2 + 2, then any intersecting family F of -subspaces of an n-dimensional vector space over GF(q) with F F F = 0 has size at most n q ( ) n + q This bound 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 Introduction Sets Let X be an n-element set and, for 0 n, let ( X ) denote the family of all subsets of X of cardinality A family F ( X ) is called intersecting if for all F, F 2 F we have F F 2 Erdős, Ko, and Rado 5 determined the maximum size of an intersecting family, and introduced the so-called shifting technique the electronic journal of combinatorics 7 (200), #R7

2 Theorem (Erdős-Ko-Rado) Suppose F ( X ) is intersecting and n 2 Then F ( { ( n ) Excepting the case n = 2, equality holds only if F = F X ) } : x F for some x X For any family F ( X ), the covering number τ(f) is the minimum size of a set that meets all F F Theorem shows that if F ( X ) is an intersecting family of maximum size and n > 2, then τ(f) = Hilton and Milner 5 determined the maximum size of an intersecting family with τ(f) 2 Later, Franl and Füredi 9 gave an elegant proof of Theorem 2 using the shifting technique Theorem 2 (Hilton-Milner) Let F ( X ) be an intersecting family with 2, n 2 +, and τ(f) 2 Then F ( ) ( n n ) + Equality holds only if (i) F = {F } {G ( X ) : x G, F G } for some -subset F and x X \ F (ii) F = {F ( X 3) : F S 2} for some 3-subset S if = 3 2 ector spaces Theorem and Theorem 2 have natural extensions to vector spaces We let always denote an n-dimensional vector space over the finite field GF(q) For Z +, we write to denote the family of all -dimensional subspaces of For a, q Z+, define the Gaussian binomial coefficient by a := q a i q q i 0 i< A simple counting argument shows that the size of q is n From now on, we will q omit the subscript q If two subspaces of intersect in the zero subspace, then we say they are disjoint or that they trivially intersect; otherwise we say the subspaces non-trivially intersect A family F is called intersecting if any two -spaces in F non-trivially intersect The maximum size of an intersecting family of -spaces was first determined by Hsieh 6 For alternate proofs of Theorem 3, see 4 and We remar that there is as yet no analog of the shifting technique for vector spaces Theorem 3 (Hsieh) Suppose F is intersecting and n 2 Then F n Equality holds if and only if F = { F : v F } for some one-dimensional subspace v, unless n = 2 Let the covering number τ(f) of a family F be defined as the minimum dimension of a subspace of that intersects all elements of F nontrivially Theorem 3 shows that, as in the set case, if F is a maximum intersecting family of -spaces, then τ(f) = Families satisfying τ(f) = are nown as point-pencils the electronic journal of combinatorics 7 (200), #R7 2

3 In this paper, we will extend Theorem 2 to vector spaces, and determine the maximum size of an intersecting family F with τ(f) 2 For two subspaces S, T, we let S +T denote their linear span We observe that for a fixed -subspace E and a -subspace U with E U, the family F E,U = {U} {W : E W, dim(w U) } is not maximal as we can add all subspaces in E+U that are not in FE,U We will say that F is an HM-type family if F = { W } : E W, dim(w U) E+U for some E and U with E U If F is an HM-type family, then its size is n n F = f(n,, q) := q ( ) + q () The main result of the paper is the following theorem Theorem 4 Suppose 3, and either q 3 and n 2 +, or q = 2 and n 2 +2 For any intersecting family F with τ(f) 2, we have F f(n,, q) (with f(n,, q) as in ()) Equality holds only if (i) F is an HM-type family, (ii) F = F 3 = {F : dim(s F) 2} for some S 3 if = 3 Furthermore, if 4, then there exists an ǫ > 0 (independent of n,, q) such that if F ( ǫ)f(n,, q), then F is a subfamily of an HM-type family If = 2, then a maximal intersecting family F of -spaces with τ(f) > is the family of all 2-subspaces of a 3-subspace, and the conclusion of the theorem holds After proving Theorem 4 in Section 2, we apply this result to determine the chromatic number of q-kneser graphs The vertex set of the q-kneser graph qk n: is Two vertices of qk n: are adjacent if and only if the corresponding -subspaces are disjoint In 3, the chromatic number of the q-kneser graph qk n:2 is determined, and the minimum colorings are characterized In 8, the chromatic number of the q-kneser graph is determined in general for q > q In Section 4, we prove the following theorem Theorem 5 If 3, and either q 3 and n 2 +, or q = 2 and n 2 + 2, then the chromatic number of the q-kneser graph is χ(qk n: ) = n + Moreover, each color class of a minimum coloring is a point-pencil and the points determining a color are the points of an (n + )-dimensional subspace In Section 5, we prove the non-uniform version of the Erdős-Ko-Rado theorem Theorem 6 Let F be an intersecting family of subspaces of the electronic journal of combinatorics 7 (200), #R7 3

4 (i) If n is even, then F n n/2 + n i>n/2 i (ii) If n is odd, then F i>n/2 n i For even n, equality holds only if F = >n/2 {F n/2 : E F } for some E, or if F = >n/2 U for some U For odd n, equality holds only if F = n/2 n >n/2 Note that Theorem 6 follows from the profile polytope of intersecting families which was determined implicitly by Bey and explicitly by Gerbner and Patós 2, but the proof we present in Section 5 is simple and direct 2 Proof of Theorem 4 This section contains the proof of Theorem 4 which we divide into two cases 2 The case τ(f) = 2 For any A and F, let FA = {F F : A F } First, let us state some easy technical lemmas Lemma 2 Let a 0 and n a + and q 2 Then n a n a < a (q )q n 2 a Proof The inequality to be proved simplifies to (q a )(q )q n 2 < q n a Lemma 22 Let E If E L, where L is an l-subspace, then the number of -subspaces of containing E and intersecting L is at least l n 2 2 q l n (with equality for l = 2), and at most l n 2 2 Proof The -spaces containing E and intersecting L in a -dimensional space are counted exactly once in the first term Those subspaces that intersect L in a 2-dimensional space are counted 2 = q+ times in the first term and q times in the second term, thus once overall If a subspace intersects L in a subspace of dimension i 3, then it is counted i times in the first term and q i 2 times in the second term, and hence a negative number of times overall Our next lemma gives bounds on the size of an HM-type family that are easier to wor with than the precise formula mentioned in the introduction Lemma 23 Let n 2 +, 3 and q 2 If F is an HM-type family, then ( ) n 2 q 3 q 2 < n 2 2 q n f(n,, q) = F n 2 2 the electronic journal of combinatorics 7 (200), #R7 4

5 Proof Since q 2 = ( )/(q + ) and n 2 +, the first inequality follows from Lemma 2 Let F be the HM-type family defined by the -space E and the -space U Then F contains all -subspaces of containing E and intersecting U, so that the second inequality follows from Lemma 22 For the last inequality, Lemma 22 almost suffices, but we also have to count the -subspaces of E+U that do not contain E Each ( )-subspace W of U is contained in q + such subspaces, one of which is E + W On the other hand, E + W was counted at least q + times since 3 This proves the last inequality Lemma 24 If a subspace S does not intersect each element of F, then there is a subspace T > S with dim T = dim S + and F T F S / Proof There is an F F such that S F = 0 Average over all T = S + E where E is a -subspace of F Lemma 25 If an s-dimensional subspace S does not intersect each element of F, then F S n s s Proof There is an (s + )-space T with n s s FT F S / Corollary 26 Let F be an intersecting family with τ(f) s Then for any i-space L with i s we have F L s i n s s Proof If i = s, then clearly F L n s s If i < s, then there exists an F F such that F L = 0; now apply Lemma 24 s i times Before proving the q-analogue of the Hilton-Milner theorem, we describe the essential part of maximal intersecting families F with τ(f) = 2 Proposition 27 Let n 2 and let F be a maximal intersecting family with τ(f) = 2 Define T to be the family of 2-spaces of that intersect all subspaces in F One of the following three possibilities holds: (i) T = and n 2 2 < F < n 2 ( 2 + (q + ) ) n 3 3 ; (ii) T >, τ(t ) =, and there is an (l + )-space W (with 2 l ) and a -space E W so that T = {M : E M W, dim M = 2} In this case, l n 2 2 q l n F l n ( l ) n q l n l l For l = 2, the upper bound can be strengthened to F (q + ) n 2 2 q n ( 2 ) n q 2 n 3 3 ; (iii) T = A 2 for some 3-subspace A and F = {U : dim(u A) 2} In this case, F = (q 2 + q + )( n 2 2 n 3 3 ) + n 3 3 the electronic journal of combinatorics 7 (200), #R7 5

6 Proof Let F be a maximal intersecting family with τ(f) = 2 By maximality, F contains all -spaces containing a T T Since n 2 and 2, two disjoint elements of T would be contained in disjoint elements of F, which is impossible Hence, T is intersecting Observe that if A, B T and A B < C < A + B, then C T As an intersecting family of 2-spaces is either a family of 2-spaces containing some fixed -space E or a family of 2-subspaces of a 3-space, we get the following: ( ): T is either a family of all 2-subspaces containing some fixed -space E that lie in some fixed (l + )-space with l, or T is the family of all 2-subspaces of a 3-space (i) : If T =, then let S denote the only 2-space in T and let E S be any -space Since τ(f) >, there exists an F F with E F, for which we must have dim(f S) = As S is the only element of T, for any -subspace E of F different from F S, we have F E+E n 3 3 by Lemma 25 Hence the number of subspaces n 3 containing E but not containing S is at most ( ) 3 This gives the upper bound (ii) : Assume that τ(t ) = and T > By ( ), T is the set of 2-spaces in an (l+)- space W (with l 2) containing some fixed -space E Every F F \ F E intersects W in a hyperplane Let L be a hyperplane in W not on E Then F contains all -spaces on E that intersect L Hence the lower bound and the first term in the upper bound come from Lemma 22 The second term comes from using Lemma 25 to count the -spaces of F that contain E and intersect a given F F (not containing E) in a point of F \ W If l 3, then there are q l hyperplanes in W not containing E and there are n l l -spaces through such a hyperplane; this gives the last term For l = 2, we use the tight lower bound in Lemma 22 to count the number of -spaces on E that intersect L There are q 2 hyperplanes in W, and they cannot be in T, so Lemma 25 gives the bound (iii) : This is immediate Corollary 28 Let F be a maximal intersecting family with τ(f) = 2 Suppose q 3 and n 2 +, or q = 2 and n If F is at least as large as an HM-type family and > 3, then F is an HM-type family If = 3, then F is an HM-type family or an F 3 -type family There exists an ǫ > 0 (independent of n,, q) such that if 4 and F is at least ( ǫ) times the size of an HM-type family, then F is an HM-type family Proof Apply Proposition 27 Note that the HM-type families are precisely those from case (ii) with l = Let n = 2 + r where r We have F / n 2 2 < + q+ (q )q in case (i) of r Proposition 27 by Lemma 2 We have F / n 2 2 < ( + ) q (q )q r + q 2 in case (ii) (q )q r when l < In both cases, for q 3 and 3, or q = 2, 4, and r 2, this is less than ( ǫ) times the lower bound on the size of an HM-type family given in Lemma 23 Using the stronger estimate in Lemma 23, we find the same conclusion for q = 2, = 3, and r 2 the electronic journal of combinatorics 7 (200), #R7 6

7 n 2 n 3 In case (iii), F 3 = q 3 q q 3 For 4, this is much smaller than the size of the HM-type families For = 3, the two families have the same size Proposition 29 Suppose that 3 and n 2 Let F be an intersecting family with τ(f) 2 Let 3 l If there is an l-space that intersects each F F and then there is an (l )-space that intersects each F F F > l l n l l, (22) Proof By averaging, there is a -space P with F P F / l If τ(f) = l, then by Corollary 26, F l l n l l, contradicting the hypothesis Corollary 20 Suppose 3 and either q 3 and n 2+, or q = 2 and n 2+2 Let F be an intersecting family with τ(f) 2 If F > 3 2 n 3 3, then τ(f) = 2; that is, F is contained in one of the systems in Proposition 27, which satisfy the bound on F Proof By Lemma 2 and the conditions on n and q, the right hand side of (22) decreases as l increases, where 3 l Hence, by Proposition 29, we can find a 2-space that intersects each F F Remar 2 For n 3, all systems described in Proposition 27 occur 22 The case τ(f) > 2 Suppose that F is an intersecting family and τ(f) = l > 2 We shall derive a contradiction from F f(n,, q), and even from F ( ǫ)f(n,, q) for some ǫ > 0 (independent of n,, q) 22 The case l = First consider the case l = Then F by Corollary 26 On the other hand, ( ) F n 2 ( ) q 3 q 2 > ( ) q 3 q (q )q n 2 2 by Lemma 23 and Lemma 2 If either q 3, n 2+ or q = 2, n 2+2, then either = 3, (n,, q) = (9, 4, 3), or (n,, q) = (0, 4, 2) If (n,, q) = (9, 4, 3) then f(n,, q) = , and 40 4 = , which gives a contradiction If (n,, q) = (0, 4, 2), then f(n,, q) = 537, and 5 4 = 50625, which again gives a contradiction Hence = 3 Now F ( ) n 2 q 3 q 2 gives a contradiction for n 8, so n = 7 Therefore, if we assume that n 2 + and either q 3, (n, ) (7, 3) or q = 2, n then we are not in the case l = It remains to settle the case n = 7, = l = 3, and q 3 By Lemma 24, we can choose a -space E such that F E F / 3 and a 2-space S on E such that FS F E / 3 the electronic journal of combinatorics 7 (200), #R7 7

8 Then F S > q+ since F > Pic F F disjoint from S and define H := S+F All F F S are contained in the 5-space H Since F > 5 3, there is an F0 F not contained in H If F 0 S = 0, then each F F S is contained in S +(H F 0 ); this implies F S q +, which is impossible Thus, all elements of F disjoint from S are in H Now F 0 must meet F and S, so F 0 meets H in a 2-space S 0 Since F S > q +, we can find two elements F, F 2 of F S with the property that S 0 is not contained in the 4-space F +F 2 Since any F F disjoint from S is contained in H and meets F 0, it must meet S 0 and also F and F 2 Hence the number of such F s is at most q 5 Altogether F q ; the first term comes from counting F F disjoint from S and the second term comes from counting F F on a given one-dimensional subspace E < S This contradicts F ( q 3 q ) The case l < Assume, for the moment, that there are two l-subspaces in that non-trivially intersect all F F, and that these two l-spaces meet in an m-space, where 0 m l By Corollary 26, for each -subspace P we have F P l n l l, and for each 2-subspace L we have F L l 2 n l l Consequently, F m l n l l + ( l m ) 2 l 2 n l l (23) The upper bound (23) is a quadratic in x = m and is largest at one of the extreme values x = 0 and x = l The maximum is taen at x = 0 only when l 2 > l 2 ; that is, when = l Since we assume that l <, the upper bound in (23) is largest for m = l We find F l l n l l + ( l l ) 2 l 2 n l l On the other hand, F ( q 3 q ) n 2 2 > ( ) l n l q 3 q l ((q )q n 2 ) l 2 Comparing these, and using > l, n 2 +, and n if q = 2, we find either (n,, l, q) = (9, 4, 3, 3) or q = 2, n = 2 + 2, l = 3, and 5 If (n,, l, q) = (9, 4, 3, 3) then f(n,, q) = , while the upper bound is , which is a contradiction If (n,, l, q) = (2, 5, 3, 2) then f(n,, q) = , while the upper bound is , which is a contradiction If (n,, l, q) = (0, 4, 3, 2) then f(n,, q) = 537, while the upper bound is 6205, which is a contradiction Hence, under our assumption that there are two distinct l-spaces that meet all F F, the case 2 < l < cannot occur We now assume that there is a unique l-space T that meets all F F We can pic a -space E < T such that F E F / l Now there is some F F not on E, so E is in lines such that each F FE contains at least one of these lines Suppose L is one of these lines and L does not lie in T; we can enlarge L to an l-space that still does not the electronic journal of combinatorics 7 (200), #R7 8

9 meet all elements of F, so F L l n l l by Lemma 24 and Lemma 25 If L does lie on T, we have F L l 2 n l l by Corollary 26 Hence, F l FE ( l l ( l 2 n l l ) + ( l )( l n l ) l ) On the other hand, we have F > ( ) q 3 q ((q )q n 2 ) l 2 l n l l Under our standard assumptions n 2 + and n if q = 2, this implies q = 2, n = 2 + 2, l = 3, which gives a contradiction We showed: If q 3 and n 2 + or if q = 2 and n 2 + 2, then an intersecting family F with F f(n,, q) must satisfy τ(f) 2 Together with Corollary 28, this proves Theorem 4 3 Critical families A subspace will be called a hitting subspace (and we shall say that the subspace intersects F), if it intersects each element of F The previous results just used the parameter τ, so only the hitting subspaces of smallest dimension were taen into account A more precise description is possible if we mae the intersecting system of subspaces critical Definition 3 An intersecting family F of subspaces of is critical if for any two distinct F, F F we have F F, and moreover for any hitting subspace G there is a F F with F G Lemma 32 For every non-extendable intersecting family F of -spaces there exists some critical family G such that F = {F : G G, G F } Proof Extend F to a maximal intersecting family H of subspaces of, and tae for G the minimal elements of H The following construction and result are an adaptation of the corresponding results from Erdős and Lovász 6: Construction 33 Let A,, A be subspaces of such that dim A i = i and dim(a + + A ) = ( ) + 2 Define F i = {F : Ai F, dim A j F = for j > i} Then F = F F is a critical, non-extendable, intersecting family of -spaces, and F i = i+ i+2 for i the electronic journal of combinatorics 7 (200), #R7 9

10 For subsets Erdős and Lovász proved that a critical, non-extendable, intersecting family of -sets cannot have more than members They conjectured that the above construction is best possible but this was disproved by Franl, Ota and Toushige 0 Here we prove the following analogous result Theorem 34 Let F be a critical, intersecting family of subspaces of of dimension at most Then F Proof Suppose that F > By induction on i, 0 i, we find an i-dimensional subspace A i of such that F Ai > i Indeed, since by induction FAi > and F is critical, the subspace A i is not hitting, and there is an F F disjoint from A i Now all elements of F Ai meet F, and we find A i+ > A i with F Ai+ > F Ai / For i = this is a contradiction Remar 35 For l this argument shows that there are not more than l l-spaces in F If l = 3 and τ > 2 then for the size of F the previous remar essentially gives 3 2 n 3 3, which is the bound in Corollary 20 Modifying the Erdős-Lovász construction (see Franl 7), one can get intersecting families with many l-spaces in the corresponding critical family Construction 36 Let A,,A l be subspaces with dim A =, dim A i = + i l for i 2 Define F i = {F : Ai F, dim(f A j ) for j > i} Then F F l is intersecting and the corresponding critical family has at least l+2 l-spaces For n large enough the Erdős-Ko-Rado theorem for vector spaces follows from the obvious fact that no critical, intersecting family can contain more than one -dimensional member The Hilton-Milner theorem and the stability of the systems follow from ( ) which was used to describe the intersecting systems with τ = 2 As remared above, the fact that the critical family has to contain only spaces of dimension 3 or more limits its size to O( n 3 ), if is fixed and n is large enough Stronger and more general stability theorems can be found in Franl 8 for the subset case 4 Coloring q-kneser graphs In this section, we prove Theorem 5 We will need the following result of Bose and Burton 2 and its extension by Metsch 7 Theorem 4 (Bose-Burton) If E is a family of -subspaces of such that any - subspace of contains at least one element of E, then E n + Furthermore, equality holds if and only if E = H for some (n + )-subspace H of l the electronic journal of combinatorics 7 (200), #R7 0

11 Proposition 42 (Metsch) If E is a family of n + ε -subspaces of, then the number of -subspaces of that are disjoint from all E E is at least εq ( )(n ) Proof of Theorem 5 Suppose that we have a coloring with at most n + colors Let G (the good colors) be the set of colors that are point-pencils and let B (the bad colors) be the remaining set of colors Then G + B n + Suppose B = ε > 0 By Proposition 42, the number of -spaces with a color in B is at least εq ( )(n ), so that the average size of a bad color class is at least q ( )(n ) This must be smaller than the size of a HM-type family Thus, by Lemma 23, n 2 q ( )(n ) 2 For 3 and q 3, n 2 + or q = 2, n 2 +2, this is a contradiction (The weaer form of Proposition 42, as stated in 7, suffices unless q = 2, n = 2 + 2) If B = 0, all color classes are point-pencils, and we are done by Theorem 4 5 Proof of Theorem 6 Let a + b = n, a < b and let F a = F a and Fb = F b We prove F a + F b n b (54) with equality only if F a = and F b = b Adding up (54) for n/2 < b n gives the bound on F in Theorem 6 if n is odd; adding the result of Greene and Kleitman 4 that states F n/2 n n/2 proves it for even n For the uniqueness part of Theorem 6, we only have to note that if n is even then, by results of Godsil and Newman 3, we must have F n/2 = {F n/2 : E F } or Fn/2 = U n/2 for some U for some E n Now we prove (54) Consider the bipartite graph with vertex set ( a, b ) and join A a and B b if A B = 0 Observe that Fa F b is an independent set in this graph Now, this graph is regular with degree q ab Therefore any independent set in this graph has size at most n b by König s Theorem Moreover, independent sets of size n b can only be a or b, but the former is not an intersecting family This proves (54) Acnowledgements Ameera Chowdhury thans the NSF for supporting her and the Rényi Institute for hosting her while she was an NSF-CESRI fellow during the summer of 2008 Balázs Patós s research was supported by NSF Grant #: CCF Tamás Szőnyi gratefully acnowledges the financial support of NWO, including the support of the DIAMANT and Spinoza projects He also thans the Department of Mathematics at TU/e for the warm hospitality He was partly supported by OTKA Grants T and NK the electronic journal of combinatorics 7 (200), #R7

12 References C Bey Polynomial LYM inequalities Combinatorica, 25():9 38, R C Bose and R C Burton A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes J Combin Theory, :96 04, A Chowdhury, C Godsil, and G Royle Colouring lines in projective space J Combin Theory Ser A, 3():39 52, A Chowdhury and B Patós Shadows and intersections in vector spaces J Combin Theory Ser A, to appear 5 P Erdős, C Ko, and R Rado Intersection theorems for systems of finite sets Quart J Math Oxford Ser (2), 2:33 320, 96 6 P Erdős and L Lovász Problems and results on 3-chromatic hypergraphs and some related questions In Infinite and finite sets (Colloq, Keszthely, 973; dedicated to P Erdős on his 60th birthday), ol II, pages Colloq Math Soc János Bolyai, ol 0 North-Holland, Amsterdam, P Franl On families of finite sets no two of which intersect in a singleton Bull Austral Math Soc, 7():25 34, P Franl On intersecting families of finite sets J Combin Theory Ser A, 24(2):46 6, P Franl and Z Füredi Nontrivial intersecting families J Combin Theory Ser A, 4():50 53, P Franl, K Ota, and N Toushige Covers in uniform intersecting families and a counterexample to a conjecture of Lovász J Combin Theory Ser A, 74():33 42, 996 P Franl and R M Wilson The Erdős-Ko-Rado theorem for vector spaces J Combin Theory Ser A, 43(2): , D Gerbner and B Patós Profile vectors in the lattice of subspaces Discrete Math, 309(9): , CD Godsil and MW Newman Independent sets in association schemes Combinatorica, 26(4):43 443, C Greene and DJ Kleitman Proof techniques in the theory of finite sets In Studies in combinatorics, MAA Stud Math 7, pages A J W Hilton and E C Milner Some intersection theorems for systems of finite sets Quart J Math Oxford Ser (2), 8: , WN Hsieh Intersection theorems for systems of finite vector spaces Discrete Math, 2: 6, K Metsch How many s-subspaces must miss a point set in PG(d, q) J Geom, 86(-2):54 64, T Mussche Extremal combinatorics in generalized Kneser graphs PhD thesis, Technical University Eindhoven, 2009 the electronic journal of combinatorics 7 (200), #R7 2