Two-Part Set Systems

Two-part set systems Péter L. Erdősa,∗ Dhruv Mubayic,‡ a Hungarian b Theoretical Dániel Gerbnera,† Cory Palmera,d,† Nathan Lemonsa,b,† Balázs Patkósa,§ Academy of Sciences, Alfréd Rényi Institute of Mathematics, P.O.B. 127, Budapest H-1364, Hungary Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA c University d University of Illinois at Chicago, Chicago, IL 60607, US of Illinois at Urbana-Champaign, Urbana, IL 61801, US Submitted: Oct 1, 2011; Accepted: Feb 21, 2012; Published: Mar 9, 2012 Mathematics Subject Classification: 05D05 Abstract The two part Sperner theorem of Katona and Kleitman states that if X is an n-element set with partition X1 ∪ X2 , and F is a family of subsets of X such that no two sets A, B
∈ F satisfy A ⊂ B (or B ⊂ A) and A ∩ Xi = B ∩ Xi for some i, n then |F| ≤ ⌊n/2⌋ . We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfy various combinations of these properties on one or both parts X1 , X2 . Along the way, we prove the following new result which may be of independent interest: let F, G be intersecting families of subsets of an n-element set that are additionally crossSperner, meaning that if A ∈ F and B ∈ G, then A 6⊂ B and B 6⊂ A. Then |F| + |G| ≤ 2n−1 and there are exponentially many examples showing that this bound is tight. Keywords: extremal set theory, Sperner, intersecting ∗ Research supported in part by the Hungarian NSF, under contract NK 78439 and K 68262. email: [email protected] † Research supported in part by the Hungarian NSF, under contract NK 78439. email: [email protected], [email protected], [email protected] ‡ Research supported in part by NSF grant DMS-0969092. email: [email protected] § Research supported by Hungarian NSF, under contract PD-83586, and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. email: [email protected] the electronic journal of combinatorics 19 (2012), #P52 1 1 Introduction Let X be a finite set and let 2X be the system of all subsets of X. The basic problem in extremal set theory is to determine the maximum size that a set system F ⊆ 2X can have provided F satisfies a prescribed property. The prototypes of investigated properties are the intersecting and Sperner properties. A set system F is intersecting if F1 ∩ F2 6= ∅ for any pair F1 , F2 ∈ F and a set system F is Sperner if there do not exist two distinct sets F1 , F2 ∈ F such that F1 ⊂ F2 . The celebrated theorems of Erdős, Ko, Rado [4] and of Sperner [13] determine the largest size that a uniform intersecting set system and Sperner system can have. Both theorems have many applications and generalizations. One such generalization of the Sperner property is the so called more part Sperner property. In this case, the underlying set X is partitioned into m subsets X1 , . . . , Xm and the system F ⊂ 2X is said to be m-part Sperner if for any pair F1 , F2 ∈ F with F1 ⊂ F2 there exist at least two indices 1 ≤ i1 < i2 ≤ m such that F1 ∩ Xij 6= F2 ∩ Xij holds for j = 1, 2. Systems with this property were first considered in [9, 11]; for a survey of recent results see [2]. In this paper we will consider analogous problems for intersection properties and also some mixed more part properties in the case when m equals 2. All maximum size 2part Sperner set systems were described by P.L. Erdős and G.O.H. Katona in [5, 6]. To rephrase the 2-part Sperner property it is convenient to introduce the following set systems of traces: for any A ⊆ X1 and B ⊆ X2 let FA = {F ∩ X2 : F ∈ F, F ∩ X1 = A}, FB = {F ∩ X1 : F ∈ F, F ∩ X2 = B}. Also, for any F ∈ F we will call F ∩ X1 and F ∩ X2 the traces of F on X1 and X2 . One can easily see that a set system F is 2-part Sperner with respect to the partition X = X1 ∪ X2 if and only if for any subset A ⊆ X1 or B ⊆ X2 the set systems FA and FB possess the Sperner property. Having this equivalence in mind, it is natural to introduce the following three definitions where we always assume that the underlying set X is partitioned into two sets X1 and X2 : Definition 1. (i) a set system F ⊆ 2X is 2-part intersecting (a 2I-system for short) if for any subset A of X1 (and for any subset B of X2 ) the trace system FA on X2 (and the trace system FB on X1 ) is intersecting, (ii) a set system F ⊆ 2X is 2-part intersecting, 2-part Sperner (a 2I2S-system for short) if for any subset A of X1 and for any subset B of X2 the trace systems FA on X2 and FB on X1 are intersecting and Sperner, (iii) a set system F ⊆ 2X is 1-part intersecting, 1-part Sperner (a 1I1S-system for short) if there exists no pair of distinct sets F1 , F2 in F such that the traces of F1 , F2 are disjoint at one of the parts and are in containment at the other. We will address the problem of finding the maximum possible size of a set system possessing the properties above. Some of our bounds will apply regardless of the sizes of the parts in the 2-partition and some will only apply to special cases. We will be mostly interested when |X1 | = |X2 |. Clearly, for any 2-part set system F we have P in the case P |F| = A⊆X1 |FA | = B⊆X2 |FB |. As any intersecting system of subsets of X1 has size the electronic journal of combinatorics 19 (2012), #P52 2 at most 2|X1 |−1 , it follows that any 2I-system has size at most 2|X2 | 2|X1 |−1 = 2|X|−1 . In Section 2 we will prove the following theorem. Theorem 2. Let F ⊆ 2X be a 2-part intersecting system of maximum size. If the 2partition X = X1 ∪ X2 is non-trivial (i.e. X1 6= ∅, X2 6= ∅), then the following inequality holds: 3 |F| ≤ 2|X| . 8 The bound is best possible if X1 or X2 is a singleton.
Moreover, if |X1 | = |X2 |, then there exists a 2-part intersecting system of size 31 2|X| + 2 . The rest of Section 2 is devoted to 2I2S systems. We prove the following result. Theorem 3. Let F ⊆ 2X
be a 2-part intersecting, 2-part Sperner system of maximum |X| holds. This bound is asymptotically sharp as long as |X1 | = size. Then |F| ≤ ⌈|X|/2⌉
|X| with o(|X2 |1/2 ). If |X1 | = |X2 | holds, then there exists a 2I2S system of size c ⌈|X|/2⌉ c > 2/3. The main result of the paper is proved in Section 3. We determine the maximum size of a 1-part intersecting 1-part Sperner set system. Theorem 4. Let F be a maximum size 1-part intersecting, 1-part Sperner set system. Then |F| = 2|X|−2 . 2 2I- and 2I2S-systems In this section we consider two-part intersecting and two-part intersecting, two-part Sperner set systems. We first consider a general construction that produces large families with these properties. Let A1 , . . . , Am and B1 , . . . , Bm be partitions of 2X1 and 2X2 into disjoint intersecting (or intersecting, Sperner) systems some of which may possibly be empty. Then the set system F := ∪m i=1 Ai × Bi = {A ∪ B : A ∈ Ai , B ∈ Bi for some 1 ≤ i ≤ m} is a 2I- (2I2S)-system by definition. Fact 5. Let 0 ≤ x1 ≤ · · · ≤ xn , 0 ≤ y1 ≤ · · · ≤ yn be real numbers and π be a permutation of the first n integers. Then we have the following inequalities: ( n ) n n n X X X X 2 2 xi , yi . xi yπ(i) ≤ xi yi ≤ max i=1 i=1 i=1 i=1 Thus to maximize the size of a family obtained through the general construction one should enumerate the Ai ’s and the Bi ’s in decreasing order according to their size. Moreover,Pif |X1 | = |X2 |, then it is enough to consider partitions A1 , . . . , Am of 2X1 and 2 the sum m i=1 |Ai | . the electronic journal of combinatorics 19 (2012), #P52 3 2.1 Two-part intersecting systems In this subsection we prove Theorem 2. In the proof we use the following theorem of Kleitman [10]. Theorem 6 (Kleitman [10]). Let F1 , . . . , Fm ⊆ 2[n] be intersecting set systems. Then |F1 ∪ · · · ∪ Fm | ≤ 2n − 2n−m . Proof of Theorem 2. For any subset A of X1 let Ā denote its complement X1 \ A. By definition, both FA and FĀ are intersecting. Also, these set systems are disjoint as B ∈ FA ∩ FĀ implies B ∪ A, B ∪ Ā ∈ F which contradicts the 2-part intersecting property of F. Thus by Theorem 6 we have |FA | + |FĀ | ≤ 2|X2 |−1 + 2|X2 |−2 . Altogether we obtain 3 |F| ≤ 2|X1 |−1 (2|X2 |−1 + 2|X2 |−2 ) = 2|X| . 8 Our best lower bounds arise from our general construction. If X1 consists of a single element x1 , then let A1 = {{x1 }}, A2 = {∅} and B1 = {B ⊂ X2 : x2 ∈ B}, B2 = {B ⊂ X2 : x2 ∈ / B, x′2 ∈ B} for two fixed elements x2 , x′2 ∈ X2 and the other Bi ’s be arbitrary while the other Ai ’s be empty. For the set system F we obtain via the general construction, we have |F| = 2|X2 |−1 + 2|X2 |−2 = 83 2|X| . Finally, let us suppose that |X1 | = |X2 | = |X|/2 and let the elements of X1 and X2 be x11 , . . . , x1m and x21 , . . . , x2m . Let us define the partition of 2X1 and 2X2 in the following way: Ai := {A ⊂ X1 \ {x11 , . . . , x1i−1 } : x1i ∈ A}, Bi := {B ⊂ X2 \ {x21 , . . . , x2i−1 } : x2i ∈ B} for all 1 ≤ i ≤ m + 1 (i.e. Am+1 = Bm+1 = {∅}). Then for the set system F arising from the general construction we have |F| = 1 + m X i=1 2|X|−2i = 2|X| + 2 . 3  Remark 7. Theorem 6 shows that the above set system for the |X1 | = |X2 | case is best possible among those that we can obtain via the general construction. Indeed, by Fact 5 we know that we have to consider of 2X1 into intersecting set systems with sizes Pm partitions s1 , s2 , . . . , sm and maximize i=1 s2i . But a partition maximizes this sum of squares if for P all 1 ≤ j ≤ m the sums ji=1 si are maximized. In the construction we use, the sums Pj i=1 si match the upper bound of Theorem 6. 2.2 Two-part intersecting, two-part Sperner systems In this subsection we consider 2I2S-systems and prove Theorem 3. To be able to use the general construction, we need to define a partition of the power set into intersecting Sperner set systems. the electronic journal of combinatorics 19 (2012), #P52 4 Construction 8. Here we give a partition of the power set of Y into intersecting Sperner systems where all levels are partitioned into minimal number of (uniform) intersecting systems (we call this canonical partition). This partition is in the form of   |Y | + 1 , . . . , |Y |; Yk , for k = 2   |Y | + 1 − 1, j = 1, . . . , |Y | − 2i + 1; Yi,j , for i = 1, . . . , 2   |Y | + 1 ∗ Yℓ , for ℓ = 0, . . . , − 1. 2
The systems Yk aren Yk . Fix an enumeration y1 , o . . . , y|Y | of the elements of Y and define

S |−2ℓ+1 Y \{y1 ,...,yj−1 } ′ ′ the systems Yi,j as Y ∈ : yj ∈ Y . Finally let Yℓ∗ = Yℓ \ |Y Yℓ,j . j=1 i We remark that the second and third types are identical to those in the corresponding Kneser construction. Note that the number of systems in the partition is quadratic in |Y | but for any ε > 0 there exists K = K(ε) such that |Y | [ k=⌈ |Y |+1 2 Yk ∪ ⌉ |Y |−2i+1 |Y |/2 [ i=|Y |/2−K|Y |1/2 [ Yi,j ∪ j=1 |Y |/2 [ ℓ=|Y |/2−K|Y Yℓ∗ ≥ (1 − ε)2|Y | . (1) |1/2 Indeed, the sets in all the Yk contain all subsets of Y of size greater than |Y |/2, and the remaining families Yi,j , Yℓ∗ contain all subsets of Y of size between |Y |/2 − K|Y |1/2 and |Y |/2. Since the number of subsets of Y of size less than |Y |/2 − K|Y |1/2 is less than ε2|Y | , the inequality in (1) follows. It is easy to see that the number of set systems in the union in (1) is at most 2K 2 |Y |. Proof of Theorem 3. The upper bound of the theorem follows from the result of Katona
|X| [9] and Kleitman [11] stating that a 2-part Sperner system has size at most ⌈|X|/2⌉ , since any 2I2S-system is 2-part Sperner. We now prove the lower bound. For i = 1, 2 let xi = |Xi |, and recall that n = |X| = x1 + x2 . First we consider the case when the size of x1 is negligible compared to the size of 1/2 x2 . Let us assume that x1 = o(x2 ). As observed above, from the canonical partition of 1 2X1 which has Θ(x21 ) families, there are m = O(x1 ) families F11 , . . . , Fm ⊂ 2X1 such that m [ Fi1 = (1 − o(1))2x1 . i=1

1/2 2 2 = is intersecting Sperner and has size (1−o(1)) xx2 /2 If i = o(x2 ) then the system x2X/2+i
n x1 1/2 (1 − o(1)) n/2 . Thus, by the general construction, we obtain the following 2I2Ssystem from these partitions:   m  [ X2 1 F= . F ∪ H : F ∈ Fi , H ∈ x2 /2 + i i=1 the electronic journal of combinatorics 19 (2012), #P52 5 By the above, |F| is equal to      X m m X n x2 1 n 1 1 |Fi | = (1 − o(1)) n . |Fi | x2 ≥ x1 (1 − o(1)) n +i 2 2 2 2 i=1 i=1 Let us consider the case x1 = x2 . We first show that the 2I2S-system F we derive
n . from the canonical partition using our general construction has size (2/3 − o(1)) ⌈n/2⌉ We then use Frankl and Füredi’s construction [7] to improve this bound by a constant factor. For sake of simplicity, assume n is divisible by 4. Then our system has size  2 X 2 n/2 n/4 n/2−2i  n/4  X X n − 1 − k 2 X n/2 2i − 1 + + , i i − 1 i i=1 k=0 i=1 i=n/4+1 where the sums belong to the three different system types in the canonical partition. We can write our system as F = F1 ∪ F2 where the first subsystem corresponds to the sets listed in the first summation, and the second one consists of the other sets. Then  2 n/2 X n/2 |F1 | = = i    n/2 X n/2 n/2 i n/2 − i i=n/4+1    2   n/2 n n = 1/2 − = (1/2 − o(1)) n/2 n/4 n/2 i=n/4+1 as n/2 i
n/2 n/2−i
is the number of those n/2-subsets of X that intersect X1 in i elements.
n and Next we prove that |F2 | ≥ (1/3 − o(1))|F1 | which implies that |F2 | ≥ 1/2−o(1) 3 n/2
n 2 thus |F| ≥ ( 3 − o(1) n/2 . We consider those members of F2 which intersect X1 in i elements (and then intersect X2 in i elements too). We will show that, for most values of i, the number of these sets is roughly a third of the number of those members of F1 , which intersect X1 (and then X2 as well) in n/2 − i elements. We have to compare 2i − 1 i 2 n/2−2i  2  2  2 n/2 − 1 − k n/2 n/2 Si = + to = . i − 1 n/2 − i i k=0
2 ) = 1/3+o(1) for all n/4−n2/3 ≤ i ≤ n/4−log n We will be done, if we establish Si /( n/2 i as  2   X  n/2 2 X n/2 n + =o . n/2 − i n/2 − i n/2 2/3  X n/4−log n<i≤n/2 i<n/4−n n/2 i )2 = 1/3 + o(1) we need the following fact. P Fact 9. Let a1 ≥ a2 ≥ · · · ≥ ak > 0 positive reals with kℓ=1 aℓ = 1. If for some j < k we P P have aℓ = 2−ℓ + o(1) for all ℓ < j and kℓ=j aℓ = o(1), then kℓ=1 a2ℓ = 1/3 + o(1). To deduce Si /(
the electronic journal of combinatorics 19 (2012), #P52 6 All we have to do is to verify the conditions of Fact 9 to the numbers

n/2−ℓ 2i−1 rℓ = i−1
n/2 i for ℓ = 1, . . . , n/2 − 2i + 1 and rn/2−2i+2 = i
n/2 i P with j = min{n/4 − i, n1/4 } and k = n/2 − 2i + 2. First of all ℓ rℓ = 1 as these numbers correspond to the ratios of set systems in a partition. Next we show that rℓ = 2−ℓ + o(1) rℓ for all ℓ < j. Writing dℓ = rℓ−1 for 2 ≤ ℓ ≤ j − 1 and i = n/4 − m we obtain dℓ = rℓ rℓ−1 = n/2−ℓ i−1
n/2−ℓ+1 i−1
= n/2 − ℓ + i + 2 1 m − ℓ/2 + 3/2 1 = + = + O(n−1/3 ) n/2 − ℓ + 1 2 n/2 − ℓ + 1 2 and thus for ℓ < j ≤ n1/4 ℓ Y i 1 r1 = = + o(1) and rℓ = r1 dt = 2−ℓ (1 + O(jn−1/3 )) = 2−ℓ (1 + O(n−1/12 )). n/2 2 t=2 Finally, from m > log n it follows that j tends to infinity and thus P Consequently, kℓ=j rℓ = o(1). Pj ℓ=1 rℓ = 1 − o(1).
n It remains to show that we can modify our construction so that it has size (2/3+ε) n/2 for some fixed ε > 0. In order to do so we replace some of the set systems in the canonical
2 Pn/4 is a positive fraction of partition. First note that for any β > 0 the sum i=n/4−βn1/2 n/2 i Pn/4 n/2
2 . Thus we will be done if for each i with n/4 − βn1/2 ≤ i ≤ n/4 we can replace i=0 i the set systems of the canonical partition that contain i-sets with other i-uniform set
2 Pi systems H1i , H2i , . . . , Hsi i such that st=1 |Hti |2 is at least (1/3 + ε) n/2 for some positive i ε. Frankl and Füredi considered in [7] the following pair of i-uniform intersecting set systems on a base set Y : let Y be equipartitioned into Y1 ∪ Y2 and define     Y i : |Y1 ∩ G| > |Y1 |/2 , G1 = G ∈ i     Y i G2 = G ∈ \ G1 : |Y2 ∩ G| > |Y2 |/2 . i
They observed that if |Y | = 2i + o(i1/2 ), then |G1i ∪ G2i | = (1 − o(1)) |Yi | and that for any
α > 0 there exists β > 0 such that if |Y | ≤ 2i + βi1/2 , then |G1i ∪ G2i | ≥ (1 − α) |Yi | . Let us fix 0 < α < 1/6 and consider β as above. We define a modified version of the canonical partition for a given set Y . We replace the set systems Yi,j for all |Y | β |Y |1/2 ≤ i ≤ |Y2 | and j = 1, . . . , |Y | − 2i + 1 with G1i and G2i . As |G1i | + |G2i | ≥ − 2√ 2 2

n 2 i 2 i 2 )2 = 1/2 − α + α2 /2 which (1 − α) n/2 , the ratio of |G | + |G | and is at least 2( 1−α 1 2 2 i i is strictly larger than 1/3 by choice of α. the electronic journal of combinatorics 19 (2012), #P52 7 
n sets used Katona’s proof that a 2-part Sperner set system can contain at most ⌈n/2⌉ a theorem of Erdős [3] on the number of sets contained in the union of k Sperner set systems. Our proofs of Theorem 2 and Remark 7 used Theorem 6, Kleitman’s result on the size of the union of k intersecting families. It seems natural to ask how large can the union of k intersecting Sperner set systems be as the problem seems to be interesting on its own right and it might help establishing bounds on 2S2I-systems. Unfortunately, we were only able to determine the exact result in the very special case when k = 2 and n is odd. The result follows easily from the following theorem of Greene, Katona and Kleitman. Theorem 10 (Greene, Katona, Kleitman [8]). If F ⊆ 2[n] is an intersecting and Sperner set system, then the following inequality holds X F ∈F , |F |≤n/2 1 n |F |−1
+ X F ∈F , |F |>n/2 1 n |F |
≤ 1. Corollary 11. Let F, G ⊆ 2[n] be intersecting Sperner set systems and n = 2l + 1 an odd

n n and the inequality is sharp as shown by + l+2 integer. Then we have |F ∪ G| ≤ l+1

[n] [n] F = l+1 , G = l+2 . Proof. We may assume that F and G are disjoint. Let us add the inequality of Theorem 10 for both systems F and G. The bigger the number of the summands, the greater the cardinality of F, therefore we need to keep the summands as small as possible to obtain the greatest number of summands. The set size for which the summand is the smallest is l + 1 and the second smallest summand is for set sizes l and l + 2.
As by the disjointness n of the systems the number of smallest summands is at most l+1 , the result follows. 3 1-part intersecting, 1-part Sperner systems In this section we study 1-part Sperner 1-part intersecting set systems and prove Theorem 4. In order to prove the result we need a further definition. We say that the set systems F and G are intersecting, cross-Sperner if both F and G are intersecting and there is no F ∈ F, G ∈ G with F ⊂ G or G ⊂ F . We will prove the following theorem which can be of independent interest. Theorem 12. Let F, G ⊂ 2[n] be a pair of cross-Sperner, intersecting set systems. Then we have |F| + |G| ≤ 2n−1 and this bound is best possible. One of our main tools will be the following special case of the Four Functions Theorem of Ahlswede and Daykin [1]. Let us write A ∧ B = {A ∩ B : A ∈ A, B ∈ B} and A ∨ B = {A ∪ B : A ∈ A, B ∈ B}. the electronic journal of combinatorics 19 (2012), #P52 8 Theorem 13 (Ahlswede-Daykin, [1]). For any pair A, B of set systems we have |A||B| ≤ |A ∧ B||A ∨ B|. The other result we will use in our argument is due to Marica and Schönheim [12] and involves the difference set system ∆(F) = {F \ F ′ : F, F ′ ∈ F}. Theorem 14 (Marica – Schönheim [12]). For any set system F we have |∆(F)| ≥ |F|. Corollary 15. Let D be a downward closed set system and let F be an intersecting subsystem of D. Then the inequality 2|F| ≤ |D| holds. Proof. As D is downward closed and F ⊂ D, it follows that ∆(F) ⊂ D. Furthermore, as F is intersecting, we have F ∩ ∆(F) = ∅ and thus we are done by Theorem 14. Proof of Theorem 12. Let us begin with defining the following four set systems U = {U ⊆ [n] : ∃H ∈ F ∪ G such that H ⊆ U }, U ′ = U \ (F ∪ G), D = {D ⊆ [n] : ∃H ∈ F ∪ G such that D ⊆ H}, D′ = D \ (F ∪ G). Clearly, D′′ = {D′ : ∃F ∈ F such that D′ ⊂ F } is downward closed (and, by definition, F ⊂ D′′ ), hence by Corollary 15 we have 2|F| ≤ |D′′ |. Moreover by the cross-Sperner property, we have (D′′ \ F) ∩ G = ∅, and therefore we have D′′ \ F ⊂ D′ . Consequently |F| ≤ |D′ | and, by symmetry, |G| ≤ |D′ | also holds. Note that F ∧ G ⊂ D′ . Indeed, F ∩ G ∈ D by definition and F ∩ G ∈ F (or F ∩ G ∈ G) would contradict the cross-Sperner property. Similarly, we obtain that F ∨ G ⊂ U ′ and it is easy to see that the cross-Sperner property implies that U ′ ∩ D′ = ∅ and thus the four systems F, G, U ′ , D′ are pairwise disjoint. Now suppose as a contradiction that |F| + |G| > 2n−1 and thus |U ′ | + |D′ | < 2n−1 . By |F|, |G| ≤ |D′ | we obtain that |U ′ | < |F|, |G| and thus using Theorem 13 we have |U ′ ||D′ | < |F||G| ≤ |F ∧ G||F ∨ G| ≤ |U ′ ||D′ |, a contradiction. Finally, let us mention some pairs of set systems for which the sum of their sizes equals 2n−1 . Any maximum intersecting system F with G the empty set system is extremal, just as the pair F1 = {F ⊂ [n] : 1 ∈ F, 2 ∈ / F }, G1 = {G ⊂ [n] : 1 ∈ / G, 2 ∈ G}. Furthermore, for any k ≥ n/2 the pair Fk = {F ⊂ [n] : 1 ∈ F, |F | ≤ k}, Gk = {G ⊂ [n] : 1 ∈ / G, |G| ≥ k} has the required property, too. Proof of Theorem 4. First let us consider any pair of maximal intersecting systems A ⊆ 2X1 , B ⊆ 2X2 . Clearly, the set system F = A × B is a 1I1S-system as any pair of sets F1 , F2 ∈ F intersect both in X1 and in X2 . This shows that a maximum 1I1S-system contains at least 2|X|−2 sets. To obtain the upper bound of the theorem let F be any 1I1S-system. For any A ⊆ X1 let Ā denote X1 \ A. By definition, both FA and FĀ are intersecting systems, and no element of the first can contain any element of the second (and vice versa). In other words they form a pair of intersecting, cross-Sperner systems. Due to Theorem 12 we have |FA | + |FĀ | ≤ 2|X2 |−1 . The number of pairs A, Ā is 2|X1 |−1 therefore we have |F| ≤ 2|X|−2 . the electronic journal of combinatorics 19 (2012), #P52 9 References [1] R. Ahlswede, D. Daykin, An inequality for the weights of two families of sets, their unions and intersections, Probability Theory and Related Fields, 43 (1978), 183–185. [2] H. Aydinian, É. Czabarka, P.L. Erdős, L.A. Székely, A tour of M-part L-Sperner families, J. Comb. Theory Ser. A, 118 (2011), 702–725. [3] P. Erdős, On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc., 51 (1945), 898–902. [4] P. Erdős, C. Ko, R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford, 12 (1961), 313–318. [5] P.L. Erdős, G.O.H. Katona, Convex hulls of more-part Sperner families, Graphs and Combinatorics 2 (1986), 123–134. [6] P.L. Erdős, G.O.H. Katona, All maximum 2-part Sperner families, J. Comb. Theory Ser. A, 43 (1986), 58–69. [7] P. Frankl and Z. Füredi, Extremal problems concerning Kneser graphs, J. Comb. Theory, Ser. B, 40 (1986), 270–284. [8] C. Greene, G.O.H. Katona, D.J. Kleitman, Extensions of the Erdős-Ko-Rado theorem, SIAM 55 (1976) 1–8. [9] G.O.H. Katona, On a conjecture of Erdős and a stronger form of Sperner’s theorem, Studia Sci. Math. Hung. 1 (1966), 59–63. [10] D.J. Kleitman, Families of non-disjoint subsets, J. Comb. Theory, 1 1966, 153-155. [11] D.J. Kleitman, On a lemma of Littlewood and Offord on the distribution of certain sums, Math. Z. 90 (1965), 251–259. [12] J. Marica, J. Schönheim, Differences of sets and a problem of Graham, Can. Math. Bull. 12 (1969), 635–637. [13] E. Sperner, Ein Satz über Untermenge einer endlichen Menge, Math Z., 27 (1928) 544–548. the electronic journal of combinatorics 19 (2012), #P52 10