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Some Intersection Theorems for Ordered Sets and Graphs

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IOURNAL OF COMBINATORIAL THEORY, Series A 43, 23-37 (1986) Some Intersection Theorems for Ordered Sets and Graphs F. R. K. CHUNG* AND R. L. GRAHAM AT&T Bell Laboratories, Murray Hill, New Jersey 07974 and *Bell Communications Research, Morristown, New Jersey P. FRANKL C.N.R.S., Paris, France AND J. B. SHEARER' Universify of California, Berkeley, California Communicated by the Managing Editors Received May 22, 1984 A classical topic in combinatorics is the study of problems of the following type: What are the maximum families F of subsets of a finite set with the property that the intersection of any two sets in the family satisfies some specified condition? Typical restrictions on the intersections F n F of any F and F’ in F are: (i) FnF’# 0, where all FEF have k elements (Erdos, Ko, and Rado (1961)). (ii) IFn F’I > j (Katona (1964)). In this paper, we consider the following general question: For a given family B of subsets of [n] = { 1, 2,..., n}, what is the largest family F of subsets of [n] satsifying F,F’EF-FnFzB for some BE B. Of particular interest are those B for which the maximum families consist of so- called “kernel systems,” i.e., the family of all supersets of some fixed set in B. For example, we show that the set of all (cyclic) translates of a block of consecutive integers in [n] is such a family. It turns out rather unexpectedly that many of the results we obtain here depend strongly on properties of the well-known entropy function (from information theory). r Current address: IBM Research, PO Box 218, Yorktown Heights, New York 10598. 23 0097-3165186 $3.00 24 CHUNG ET AL. I. INTRODUCTION A classical topic in combinatorics is the study of questions of the follow- ing type: What are the maximum families F of subsets of a finite set with the property that the intersection of any two sets in the family satisfies some specified condition? Typical restrictions on the intersections based on F and F’ in F are: (i) Fn F’ # @, where F denotes the complement of F [ 163; (ii) Fn F’ # 0, where all FE F have k elements [3]; (iii) JFnF’( aj [S). Good surveys of our current state of knowledge in this area can be found in [6, 7, 9, 173, in addition to the results in [S, 12, 13, 14, 181. In this note we investigate the following question: For a given family B of subsets of [n] := { 1, 2,..., n}, what is the largest family F of subsets of [n] satisfying: F,F’eF+FnF’zB for some BE B. (1) In particular, let v(B) denote the cardinality of the largest family F satisfy- ing (1). An Easy Example As a prelude to the general results, we first consider a simple special case. For B = B, we take the set of all pairs {i, i + 1 >, 1 < i < n. For the family B2 we prove r(B*) = 2”-2. (2) Proof of (2): Define Si, i= 1,2, by S,:= {Jo [n]:j- i (mod 2)). (3) Observe that for all i and all BE B Sin B#/21. (4) Suppose F c 2r”’ satisfies (1). Define the induced families F(Si) by F(Sj):= {Fn Si: FE F}, i= 1,2. (5) Note that if G, G’ E F(S,) then GnG’=(FnSi)n(FnSi) for some F, F’ E F (6) =FnFnSi#O SOME INTERSECTION THEOREMS 25 since F n F’ 2 B’ for some B’ E B and by construction Si n B # Qr for every BE B. Thus, for i = 1,2, F( Si) is a family of subsets of Si with the property that no two sets in F(S,) are disjoint. This implies that IF( < +. 2”” (7) since we cannot have a set X and its complement Si - X both in F(S,). Since any set FE F is determined by its intersections F n Si, i = 1, 2, then by (7) IFI<;. IS11. 21. p21 = a. p + Ial = 2-2. (8) hand for the family F’ given by F'"=;X::z]: (tt2hyrcX), we’have FnF’c{l,2}~B for all F, F’ E F’ and IF’1 = 2”-*. This proves (2). 1 Note that the content of (2) is just that no family satisfying (1) for B, can have more sets than can be achieved in a trivial way, i.e., by taking all subsets of [n] containing a fixed B, E B. In general, we call such a family a kernel system with kernel B,. Of course, (2) does not imply that every maximum family F is a kernel system. In what follows, we will be especially interested in those families B for which u(B) is attained by kernel systems. This seems to be true, for exam- ple, for any family B formed by taking the (cyclic) translates of a fixed set in [n] (although we do not prove this). II. PARTITIONS OF [n] Although we study set intersections here, it is sometimes useful to con- sider the following variation of set intersection, namely, the complement of the symmetric difference of two sets, defined for X, YE [n] by XV Y:=(Xn Y)u(Xn P)=XA Y where X= [n] - X. For a given family B of subsets of [n], let C(B) denote the cardinality of the largest family F satisfying F, F’EF~FVFZB for some B E B. Obviously v(B) < C(B). 26 CHUNG ET AL. Slightly less obvious is the following. FACT 1. u(B) = C(B) for all B. (9) Sketch of prooj Assume F is a maximum V-family for B, i.e., IF/ = t?(B). Select, if possible, some element t E [n] so that for some FE F, Fu (t} & F. Replace a/Z such FE F (simultaneously) by Fu (t}, forming a new family F’. It is easy to check that F’ is also a V-family for B, and JF’J = IFI. Continue this process as long as possible, finally forming the family F*, which has the property that for any FEF’, if t $ F then Fu (t> E F*. Thus, F* is an upper ideal in the lattice of subsets 2[“‘, i.e., [n] 2 G 1 FE F* implies G E F *. It now follows easily that F* is in fact an n-family for B, i.e., F, F’ E F* implies Fn F’ 2 B for some BE B. Since JF*l = (FI =6(B) then we have u(B)2 (B) which implies (9). l THEOREM 1. Suppose [n] = S, u ... u S, is a partition of [n] into k non-empty subsets. For XC [n], define f(X) = (i: Sin X#@). Let B be a family of subsets of [n] and define B* = (AX): XE B} c 2ck1. Then we have u(B) < u(B*) 2”-k. (10) Proof By Fact 1, it is enough to prove v(B) 6 v(B*) 2”-k. (10’) Let F be a V-family for B, i.e., F, F’ E F implies F V F’ 2 B for some BE B. Also, let W denote the subspace of 2 rnl (considered as an n-dimensional vector space under the operation A) generated by the Sj. Partition Zc”’ into cosets Ci A W, 1 f id 2”- k. It will suffice to show that each coset CA W contains at most i$B*) elements of F. Since (X A C) V (Y A C) =X V Y, it suffices to prove that W contains at most C(B*) elements of F. Note that f is a one-to-one map of W to 2ck1 and it is easily checked that f(X V Y) = f(X) V f ( Y). Hence, for F, I;’ E Fn W, we have j-(F)Vf(F’)=f(FVF’)?f(B)EB* for some BE B. Therefore, W contains at most iT(B*) elements of F and Theorem 1 is proved. 1 As an immediate consequence of Theorem 1, we have the following result, which has also been obtained independently by Faudree, Schelp, and Sbs [4]. SOME INTERSECTION THEOREMS 27 THEOREM 2. Suppose [n] = S, u ... u Sk is a partition of [n] into k non-empty sets, and B G 2[“’ is a family with the property that for somej, 16 j f k, each 3 E B intersects at least j of the Si, 1 ,< i < k. Then u(B) < 2”-kg(k, j) (11) where if k+j=2u, if k+ j=2v- 1. Proof. By Fact 1 and Theorem 1 we have u(B) d u(B*) 2”- k, Since B* is a family of subsets of [k] each containing at least j elements, then a result of Kleitman [lo] (also see Ahlswede and Katona [l]) implies u(B* ) < g(k, j). This proves Theorem 2. B In order to apply Theorem 2 to a particular family B, we need to choose a suitable partition [In] = Ur=, Si (which determines some maximal value of j associated with it). It is always possible to use trivial partitions and indeed, these are sometimes optimal. For example, for [n] = S, we have k = 1, j= 1, g(k, j) = 1, and so, v(B)d2”-’ for any family B (which does not contain $3). Of course, for B = ( { 1 } }, for example, the family F = {Xc [n]: 1 E X} shows that this bound can be achieved. On the other hand, suppose we take for B the family of all j-element sub- sets of [n].
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