A DEGREE VERSION OF THE HILTON MILNER THEOREM

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1 A DEGREE VERSION OF THE HILTON MILNER THEOREM PETER FRANKL, JIE HAN, HAO HUANG, AND YI ZHAO Abstract An intersecting family of sets is trivial if all of its members share a common element Hilton and Milner proved a strong stability result for the celebrated Erdős Ko Rado theorem: when n > 2, every non-trivial intersecting family of -subsets of has at most ( ( 1 n members One extremal family HM n, consists of a -set S and all -subsets of containing a fixed element x S and at least one element of S We prove a degree version of the Hilton Milner theorem: if n = Ω( 2 and F is a non-trivial intersecting family of -subsets of, then δ(f δ(hm n, where δ(f denotes the minimum (vertex degree of F Our proof uses several fundamental results in extremal set theory, the concept of ernels, and a new variant of the Erdős Ko Rado theorem 1 Introduction A family F of sets is called intersecting if A B for all A, B F A fundamental problem in extremal set theory is to study the properties of intersecting families For positive integers, n, let = {1, 2,, n} and ( V denote the family of all -element subsets (-subsets of V We call a family on V -uniform if it is a subfamily of ( V A full star is a family that consists of all the -subsets of that contains a fixed element We call an intersecting family F trivial if it is a subfamily of a full star The celebrated Erdős Ko Rado (EKR theorem [3] states that, when n 2, every -uniform intersecting family on has at most ( 1 members, and the full star shows that the bound ( 1 is best possible Hilton and Milner [14] proved the uniqueness of the extremal family in a stronger sense: if n > 2, every non-trivial intersecting family of -subsets of has at most ( ( 1 n members It is easy to see that the equality holds for the following family, denoted by HM n,, which consists of a -set S and all -subsets of containing a fixed element x S and at least one vertex of S For more results on intersecting families, see a recent survey by Franl and Toushige [10] Given a family F and x V (F, we denote by F(x the subfamily of F consisting of all the members of F that contain x, ie, F(x := {F F : x F } Let d F (x := F(x be the degree of x Let (F := max x d F (x and δ(f := min x d F (x denote the maximum and minimum degree of F, respectively There were extremal problems in set theory that considered the maximum or minimum degree of families satisfying certain properties For example, Franl [7] extended the Hilton Milner theorem by giving sharp upper bounds on the size of intersecting families with certain maximum degree Bollobás, Dayin, and Erdős [1] studied the minimum degree version of a well-nown conjecture of Erdős [2] on matchings Huang and Zhao [15] recently proved a minimum degree version of the EKR theorem, which states that, if n > 2 and F is a -uniform intersecting family on, then δ(f ( n2 2, and the equality holds only if F is a full star This result implies the EKR theorem immediately: given a -uniform intersecting family F, by recursively deleting elements with the smallest degree until 2 elements are left, we derive that ( ( ( ( ( n 2 n n 1 F = 1 1 Franl and Toushige [11] gave a different proof of the result of [15] for n 3 Generally speaing, a minimum degree condition forces the sets of a family to be distributed somewhat evenly and thus the size of a family that is required to satisfy a property might be smaller than the one without degree condition Date: May 28, Mathematics Subject Classification 05D05 Key words and phrases Intersecting families, Hilton Milner theorem, Erdős Ko Rado theorem JH is supported by FAPESP (Proc 2013/ , 2014/ HH is partially supported by a Simons Collaboration Grant YZ is partially supported by NSF grant DMS

2 Unless the extremal family is very regular, an extremal problem under the minimum degree condition seems harder than the original extremal problem because one cannot directly apply the shifting method (a powerful tool in extremal set theory In this paper we study the minimum degree version of the Hilton Milner theorem Theorem 1 Suppose 4 and n c 2, where c = 30 for = 4, 5 and c = 4 for 6 If F ( is a non-trivial intersecting family, then δ(f δ(hm n, = ( ( n2 2 n2 2 Han and Kohayaawa [12] recently determined the maximum size of a non-trivial intersecting family that is not a subfamily of HM n,, which is ( ( 1 n1 ( 1 n Later Kostocha and Mubayi [17] determined the maximum size of a non-trivial intersecting family that is not a subfamily of HM n, or the extremal families given in [12] for sufficiently large n Furthermore, Kostocha and Mubayi [17, Theorem 8] characterized all maximal intersecting 3-uniform families F on for n 7 and F 11 Using a different approach, Polcyn and Rucińsi [18, Theorem 4] characterized all maximal intersecting 3-uniform families F on for n 7, in particular, there are fifteen such families, including the full star and HM n,3 It is straightforward to chec that all these families have minimum degree at most 3 this gives the following proposition Proposition 2 If n 7 and F ( 3 is a non-trivial intersecting family, then δ(f δ(hmn,3 = 3 In order to prove Theorem 1, we prove a new variant of the EKR theorem, which is closely related to the EKR theorem for direct products given by Franl (see Theorem 7 Theorem 3 Given integers 3, l 4, and m l, let T 1, T 2, T 3 be three disjoint l-subsets of [m] If F is a -uniform intersecting family on [m] such that every member intersects all of T 1, T 2, T 3, then F l 2( m3 3 Theorem 3 becomes trivial when l = 1 because every family F of -sets that intersect T 1, T 2, T 3 satisfies F ( m3 3 Our bound in Theorem 3 is asymptotically tight because a star with a center in T1 T 2 T 3 contains about l 2( m3 3 -sets that intersect T1, T 2, T 3 It was shown in [15] that one can derive the minimum degree version of the EKR theorem for n = Ω( 2 by using the Hilton Milner Theorem and simple averaging arguments (thus the difficulty of the result in [15] lies in deriving the tight bound n However, we can not use this naive approach to prove Theorem 1 for sufficiently large n Indeed, let F be a non-trivial intersecting family that is not a subfamily of HM n, The result of Han and Kohayaawa [12] says that F is asymptotically at most ( 1 ( n2 2, and in turn, the average degree of F is asymptotically at most (1 2 ( n3 3 Unfortunately, this is much larger than δ(hm n, ( n3 3 as is fixed and n is sufficiently large Our proof of Theorem 1 applies several fundamental results in extremal set theory as well as Theorem 3 The following is an outline of our proof Let F be a non-trivial intersecting family such that δ(f > δ(hm n, For every u, we obtain a lower bound for F \ F(u by applying the assumption on δ(f and the Franl Wilson theorem [5, 19] on the maximum size of t-intersecting families If = 4, 5, then we derive a contradiction by considering the ernel of F (a concept introduced by Franl [6] When 6, we separate two cases based on (F When (F is large, assume that F(u = (F and let F 2 := F \F(u A result of Franl [9] implies that F(u contains three edges E i := {u} T i, i [3], where T 1, T 2, T 3 are pairwise disjoint Since F 2 is intersecting and every member of F 2 meets each of T 1, T 2, T 3, Theorem 3 gives an upper bound on F 2, which contradicts the lower bound that we derived earlier When (F is small, we apply the aforementioned result of Franl [7] to obtain an upper bound on F, which contradicts the assumption on δ(f 2 Tools 21 Results that we need Given a positive integer t, a family F of sets is called t-intersecting if A B t for all A, B F A t-intersecting EKR theorem was proved in [3] for sufficiently large n Later Franl [5] (for t 15 and Wilson [19] (for all t determined the exact threshold for n 2

3 Theorem 4 [5, 19] Let n (t + 1( t + 1 and let F be a -uniform t-intersecting families on Then F ( nt t As mentioned in Section 1, Franl [7] determined the maximum possible size of an intersecting family under a maximum degree condition Theorem 5 [7] Suppose n > 2, 3 i + 1, F ( ( is intersecting If (F ( 1 ni 1, then F ( ( 1 ni ( 1 + ni i+1 Given a -uniform family F, a matching of size s is a collection of s vertex-disjoint sets of F A well-nown conjecture of Erdős [2] states that if n (s + 1 and F ( ( satisfies F > max{ n ( ns (, (s+11 }, then F contains a matching of size s + 1 Franl [9] verified this conjecture for n (2s + 1 s Theorem 6 [9] Let n (2s + 1 s and let F ( ( If F > n ( ns, then F contains a matching of size s + 1 Franl [8] proved an EKR theorem for direct products Theorem 7 [8] Suppose n = n n d and = d, where n i i are positive integers Let X 1 X d be a partition of with X i = n i, and { ( } H = F : F X i = i for i = 1,, d If n i 2 i for all i and F H is intersecting, then F H max i i n i Note that the d = 1 case of Theorem 7 is the EKR theorem 22 Kernels of intersecting families Franl introduced the concept of ernels (and called them bases for intersecting families in [6] Given F ( V, a set S V is called a cover of F if S A for all A F For example, if F is intersecting, then every member of F is a cover Given an intersecting family F, we define its ernel K as K := {S : S is a cover of F and any S S is not a cover of F} An intersecting family F is called maximal if F {G} is not intersecting for any -set G F Note that, when proving Theorem 1, we may assume that F is maximal because otherwise we can add more -sets to F such that the resulting intersecting family is still non-trivial and satisfies the minimum degree condition We observe the following fact on the ernels Fact 8 If n 2 and F ( is a maximal intersecting family, then K is also intersecting Proof Suppose there are K 1, K 2 K such that K 1 K 2 = Since n 2, we can find two disjoint -sets F 1, F 2 on such that K i F i for i = 1, 2 For i = 1, 2, since K i is a cover of F, F i intersects all members of F Since F is maximal, we derive that F 1, F 2 F This contradicts the assumption that F 1, F 2 are disjoint For i [], let K i := K ( i If an intersecting family F is non-trivial, then K1 = Below we prove an upper bound for K i, 3 i, where the i = case was given by Erdős and Lovász [4] Lemma 9 For 3 i, we have K i i In order to prove Lemma 9, We use a result of Håstad, Juna, and Pudlá [13, Lemma 34] Given a family F, the cover number of F, denoted by τ(f, is the size of the smallest cover of F Lemma 10 [13] If F is an i-uniform family with F > i, then there exists a set Y such that τ(f Y +1, where F Y := {F \ Y : F F, F Y } 3

4 Proof of Lemma 9 Suppose K i > i for some 3 i Then by Lemma 10, there exists a set Y such that τ((k i Y + 1 In particular, (K i Y is nonempty, namely, there exists K K i such that Y K By the definition of K, this implies that Y is not a cover of F, so there exists F F such that F Y = Since each member of K i is a cover of F, each of them intersects F This implies that τ((k i Y F =, a contradiction 3 Proof of Theorem 3 In this section we derive Theorem 3 from Theorem 7 Proof of Theorem 3 Let F r consist of all the subsets of F that intersect with T 1 T 2 T 3 in exactly r elements Then F = F 3 F 4 F Let X 1 = T 1, X 2 = T 2, X 3 = T 3, X 4 = [m] \ (T 1 T 2 T 3, and 1 = 2 = 3 = 1, 4 = 3 Since m l, we have 1/l ( 3/(m 3l Since l 2, we can apply Theorem 7 to conclude that ( m 3l F 3 l l = l2 ( m 3l 3 Note that a set S F 4 intersects T 1, T 2, T 3 with either 1, 1, 2 or 1, 2, 1 or 2, 1, 1 elements We partition F 4 into three subfamilies accordingly Our assumption implies 4 m 3l 2 l 1 2 We can apply Theorem 7 to each subfamily of F 4 and obtain that ( ( l m 3l F 4 3 l 2 2 ( m 3l = 3(l 1l2 2 4 l 4 Finally, for 5 r, we claim that F r l 2( ( 3l3 m3l r3 r Indeed, let X1 = T 1 T 2 T 3, X 2 = [m] \ X 1, 1 = r and 2 = r Note that X 2 = m 3l 2( r and r/(3l ( r/(m 3l If X 1 = 3l 2r, then Theorem 7 gives that F r ( 3l 1 r 1 ( m 3l r When 3l 2r, we have r 6 because l 4 Hence, ( 3l (3l 3 ( 3l 3 < r r(r 1(r 2 r 3 and the trivial bound on F r gives that ( ( 3l m 3l F r r r < l 2 ( 3l 3 r 3 18l 2 (r 1(r 2 < l 2 ( 3l 3 r 3 ( m 3l r ( ( 3l 3 3l 3 < l 2, r 3 r 3 ( m 3l r as claimed Summing up the bounds for F 3, F 4 and F r for r 5, we have because ( m3 3 F = F 3 + F 4 + F r r=5 ( ( m 3l m 3l l 2 + 3(l 1l = 3 i=0 ( m3l ( 3l3 3i i + l 2 r=5 ( 3l 3 r 3 ( m 3l r ( m 3 = l 2, 3 4

5 ( n2+t1 2 ( nt1 2 4 Proof of Theorem 1 We start with some simple estimates First, for n c 2, c 1 and 1 t 1, we have ( 2 (n 2 + t 1 (n 3 + t + 2 2t = 1 (n t 1 (n t + 2 n t + 2 2( t( 1 n t + 2 c 2 (41 c Similarly, one can show that ( ( n2 3 c1 n3 ( c 3 Second, if δ(f > n2 ( 2 n2 2, then we have F > n (( ( ( n 2 n n > n 3 ( ( (c 1n n 3 (c 1 n 2 > ( (42 c 3 c Lemma 11 Suppose 4 and n 4 2, F ( is a non-trivial intersecting family such that δ(f > δ(hm n, = ( ( n2 2 n2 2 Then for any u, (i there exists E, E ( F such that u / E E and E E = 1; (ii F \ F(u > 2 n2 2 2 Proof Given u, write F 1 = F(u and F 2 = F \ F 1 If F 2 = 1, then F HM n,, and thus δ(f δ(hm n,, a contradiction So assume that F 2 2 Let t = min E E among all distinct E, E F 2 Obviously 1 t 1, and F 2 is a t-intersecting family on [2, n] Then since n > 4 2 ( t + 1(t , we get F 2 ( nt1 t by Theorem 4 Note that there exist E, E F 2 such that E E = t Since every set in F 1 must intersect both E and E, for every x E E {u}, by the inclusion-exclusion principle, we have ( ( ( n 2 n n 2 + t 2 F 1 (x 2 + (43 Let X = \ (E E {u} and thus X = n 1 (2 t Suppose x X attains the minimum degree in F 2 among all elements of X Since F(x = F 1 (x + F 2 (x > δ(hm n,, by (43 we have ( ( n n 2 + t 2 F 2 (x > By the definition of x we get F 2 > X (( ( n n 2 + t 2 t ( n 2 + t 1 = (, X ( t t ( n 2 + t 2 where the factor t comes from the fact that every member F F 2 is counted at most t times because F E 1 t By (41 with c = 4 and 4, we get F 2 > ( ( n t 1 n t 1, 2 which, together with F 2 ( nt1 t, implies that t = 1, so (i holds Since t = 1, the first inequality above gives (ii Proof of Theorem 1 First assume that 6 and n 4 2 Suppose F ( is a non-trivial intersecting family such that δ(f > δ(hm n, = ( ( n2 2 n2 2 Suppose u attains the maximum degree of F and write F := F \ F(u If F(u > ( ( 1 n3 1, then by Theorem 6, the ( 1-uniform family {E \ {u} : E F(u} contains a matching M = {T 1, T 2, T 3 } of size 3 Every member of F must intersect each of T 1, T 2, T 3 By Theorem 3, we have F ( 1 2( n4 3 On the other hand, Lemma 11 Part (ii 5 3

6 ( implies that F > 2 n2 ( 2 2 = n2 n3 ( 2 3 > 2( 1 2 n3 3 because n 4 2 4( This gives a contradiction We thus assume that (F ( ( 1 n3 1 By Theorem 5, ( ( ( ( ( n 1 n 3 n 3 3n 2 n 2 n 2 F + = n 2 Since δ(f > ( ( n2 2 n2 2, by (42, we have F > 3 4 (( n2 2 The upper and lower bounds for F together imply < 6, a contradiction Now assume that = 4, 5 and n 30 2 Since F is intersecting, each member of F is a cover of F and thus contains as a subset a minimal cover, which is a member of the ernel K Thus F i=1 K i ( ni i We now K 1 = because F is non-trivial We observe that K 2 1 otherwise assume uv, uv K 2 (recall that K 2 is intersecting By the definition of K 2, every E F \ F(u contains both v and v so every E, E F \ F(u satisfy that E E 2, contradicting Lemma 11 Part (i By Lemma 9, ( n 2 ( n i F + i i Since n 30 2, for any 3 i, we have ( ( n i n 2 i2 = i i=3 i2 n 2 3 n 3 i + 1 n i + 1 ( n i2 Thus ( ( ( n 2 n 2 F n 2 (1 30 i i=3 On the other hand, by (42, we have F > (( n2 2 > (( n2 2 Hence, 28( < , contradicting 4 5 This completes the proof of Theorem 1 5 Concluding Remars The main question arising from our wor is whether Theorem 1 holds for all n Proposition 2 confirms this for = 3 Another question is whether the following generalization of Theorems 3 and 7 is true We say a family H of sets has the EKR property if the largest intersecting subfamily of H is trivial Conjecture 12 Suppose n = n n d and d, where n i > i 0 are integers Let X 1 X d be a partition of with X i = n i, and { ( } H := F : F X i i for i = 1,, d If n i 2 i for all i and n i > d j=1 j + i for all but at most one i [d] such that i > 0, then H has the EKR property The assumptions on n i cannot be relaxed for the following reasons If n i < 2 i for some i, then H itself is intersecting and H(x < H for any x If n i d j=1 j + i for distinct i 1, i 2 such that i1, i2 > 0, then for any x, the union of H(x and {F H : X i1 F or X i2 F } is a larger intersecting family than H(x When = d, Conjecture 12 follows from Theorem 7, in particular, the d = 1 case is the EKR theorem A recent result of Katona [16] confirms Conjecture 12 for the case d = 2 and n 1, n 2 9( min{ 1, 2 } 2 We can prove Conjecture 12 in the following case Theorem 13 Given positive integers d, 2 t 1 t 2 t d with t 2 d + 2, there exists n 0 such that the followings holds for all n n 0 If T 1,, T d are disjoint subsets of such that T i = t i for all i, then { ( } H := F : F T i 1 for i = 1,, d has the EKR property 6

7 We omit the proof of Theorem 13 here because the purpose of this paper is to prove Theorem 1 Moreover, when d = 3 and t 1 = t 2 = t 3 = 1, our n 0 is Ω( 4 so we cannot replace Theorem 3 by Theorem 13 in our main proof Nevertheless, it would be interesting to now the smallest n 0 such that Theorem 13 holds References [1] B Bollobás, D E Dayin, and P Erdős Sets of independent edges of a hypergraph Quart J Math Oxford Ser (2, 27(105:25 32, 1976 [2] P Erdős A problem on independent r-tuples Ann Univ Sci Budapest Eötvös Sect Math, 8:93 95, 1965 [3] P Erdős, C Ko, and R Rado Intersection theorems for systems of finite sets Quart J Math Oxford Ser (2, 12: , 1961 [4] P Erdős and L Lovász Problems and results on 3-chromatic hypergraphs and some related questions pages Colloq Math Soc János Bolyai, Vol 10, 1975 [5] P Franl The Erdős-Ko-Rado theorem is true for n = ct In Combinatorics (Proc Fifth Hungarian Colloq, Keszthely, 1976, Vol I, volume 18 of Colloq Math Soc János Bolyai, pages North-Holland, Amsterdam-New Yor, 1978 [6] P Franl On intersecting families of finite sets J Combin Theory Ser A, 24(2: , 1978 [7] P Franl Erdős-Ko-Rado theorem with conditions on the maximal degree J Combin Theory Ser A, 46(2: , 1987 [8] P Franl An Erdő-Ko-Rado theorem for direct products European J Combin, 17(8: , 1996 [9] P Franl Improved bounds for Erdős matching conjecture J Combin Theory Ser A, 120(5: , 2013 [10] P Franl and N Toushige Invitation to intersection problems for finite sets J Combin Theory Ser A, 144: , 2016 [11] P Franl and N Toushige A note on Huang-Zhao theorem on intersecting families with large minimum degree Discrete Math, 340(5: , 2017 [12] J Han and Y Kohayaawa The maximum size of a non-trivial intersecting uniform family that is not a subfamily of the Hilton-Milner family Proc Amer Math Soc, 145(1:73 87, 2017 [13] J Håstad, S Juna, and P Pudlá Top-down lower bounds for depth-three circuits computational complexity, 5(2:99 112, 1995 [14] A J W Hilton and E C Milner Some intersection theorems for systems of finite sets Quart J Math Oxford Ser (2, 18: , 1967 [15] H Huang and Y Zhao Degree versions of the Erdős-Ko-Rado theorem and Erdős hypergraph matching conjecture J Combin Theory Ser A, to appear [16] G O H Katona A general 2-part Erdős-Ko-Rado theorem Opuscula Math, 37(4, 2017 [17] A Kostocha and D Mubayi The structure of large intersecting families Proc Amer Math Soc, accepted, 2016 [18] J Polcyn and A Rucińsi A hierarchy of maximal intersecting triple systems Opuscula Math, 37(4, 2017 [19] R M Wilson The exact bound in the Erdős-Ko-Rado theorem Combinatorica, 4(2-3: , 1984 Alfréd Rényi Institute of Mathematics, POBox 127, H-1364 Budapest, Hungary address, Peter Franl: peterfranl@gmailcom Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, , São Paulo, Brazil address, Jie Han: jhan@imeuspbr Department of Math and CS, Emory University, Atlanta, GA address, Hao Huang: haohuang@emoryedu Department of Mathematics and Statistics, Georgia State University, Atlanta, GA address, Yi Zhao: yzhao6@gsuedu 7