The edge-diametric theorem in Hamming spaces

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1 Discrete Applied Mathematics The edge-diametric theorem in Hamming spaces Christian Bey Otto-von-Guericke-Universität Magdeburg, Institut für Algebra und Geometrie, Postfach 420, 3906 Magdeburg, Germany Received 3 April 2004; received in revised form 5 September 2004; accepted 2 November 2006 Available online 2 August 2007 Dedicated to the memory of Levon Khachatrian Abstract The maximum number of edges spanned by a subset of given diameter in a Hamming space with alphabet size at least three is determined. The binary case was solved earlier by Ahlswede and Khachatrian [A diametric theorem for edges, J. Combin. Theory Ser. A ] Elsevier B.V. All rights reserved. Keywords: Diametric problem; Hamming space; Intersection theorem. Introduction Let H n ={,...,}n be the Hamming space, i.e. H n is equipped with the Hamming distance d H, where for a = a,...,a n, b = b,...,b n H n we have d Ha, b = {i : a i = b i }. The diameter of a subset A H n is defined by diama = max{d H a, b : a,b A}. For every integer d with 0 <d<nput D,n,d={A H n : diama d}. As usual, H n is also considered as a graph V, E with vertex set V=Hn and edge set E={{a,b} :d Ha, b=}. For every subset A V let EA ={{a,b} :a,b A, d H a, b = } be the edge set induced by A. The vertex-resp. edge diametric problem in Hamming space is to determine the function resp. V,n,d:= E,n,d:= max A D,n,d A max A D,n,d EA vertex-diametric function, edge-diametric function. Clearly, these problems can be formulated in every graph. address: christian.bey@mathematik.uni-magdeburg.de X/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:0.06/j.dam

2 C. Bey / Discrete Applied Mathematics Consider for every integer t with 0 <t<nthe following subsets of H n : F i,n,t:= {a H n : Ba {,...,t + 2i} t + i}, where Ba is defined for every a = a,...,a n H n by n t 0 i 2, Ba ={i {,...,n}:a i = }. Note that F i,n,tis the product of a ball with radius i on the first t + 2i coordinates and a Hamming space on the remaining n t 2i coordinates. Thus, F i,n,t D,n,n t for all i. The vertex-diametric problem in binary Hamming spaces was solved by Kleitman: Theorem Kleitman []. V2,n,d= F d/2 2,n,n d. For larger alphabets, the complete solution of the vertex-diametric problem is due to Ahlswede and Khachatrian: Theorem 2 Ahlswede and Khachatrian [2]. Let r be the largest nonnegative integer such that 2r d and r n d / 2 are satisfied. Then V,n,d= F r,n,n d. See [2] for a list of previously obtained partial results. Equivalent versions of the above theorems in terms of intersection instead of diametry were obtained by Katona [0] Theorem and Frankl and Tokushige [9] Theorem 2. Ahlswede and Khachatrian also solved the edge-diametric problem in binary Hamming spaces: Theorem 3 Ahlswede and Khachatrian [4]. { EF0 2,n, if d = n, E2,n,d= EF d/2 2,n,n d if d n 2. Here we continue these investigations by providing a solution for the edge-diametric problem for all other alphabet sizes: Theorem 4. Let 3. Then { EF0,n,n d if d = n,n 2, E,n,d= max EF i,n,n d if d<n 2. 0 i d/2 The maximum in Theorem 4 is attained at the largest nonnegative integer i for which 2i d and + + n d d 2i / + i i are satisfied. We remark without proof that, up to permutations of the coordinates and of the alphabets in the components, the F i,n,n d are the only optimal configurations. Our proof of Theorem 4 is based on the powerful methods developed by Ahlswede and Khachatrian in [,3]. 2. Reduction to an intersection problem Let N denote the set of positive integers. For i, j N put [i, j] ={i, i +,...,j} and [i] =[,i]. Further, let 2 [i] ={A : A [i]} and ={A [i] : A =k}. [i] k

3 52 C. Bey / Discrete Applied Mathematics Consider the set M =[, n] together with the partition M = M M n where M i ={j M : j i mod n}, i =,...,n. There is a natural bijection between the Hamming space H n and the set C n ={A M : A M i = for all i =,...,n}, which maps a = a,...,a n H n to A = n i= {a i n + i} C n. Accordingly, we define for every A Cn EA := {{A,A 2 }:A,A 2 A, A A 2 =2}, where A A 2 denotes the symmetric difference of A and A 2. Recall that a system of sets A 2 N is called t-intersecting if A A 2 t for all A,A 2 A. For m N let I m, t be the set of all t-intersecting systems A 2 [m], and put I n t ={A Cn : A In, t}. The following equality is now obvious: E,n,d= max A I n n d EA. We continue with the well-known notion of left-shifted set systems [8]. Definition. For any A 2 N,anyA A and i, j N let { {i} A\{j} if i/ A, j A, {i} A\{j} / A, S i,j A = A otherwise and S i,j A ={S i,j A : A A}. A family A 2 N is called left-shifted in T where T N ifs i,j A = A for all i, j T with i<j. The shift-operations S i,j have the following easy but important properties. Lemma 5. Let A 2 N and i, j N. Then: i S i,j A C n whenever A Cn and i, j M k for some k =,...,n, ii S i,j A = A, iii S i,j A I m, t whenever A I m, t and i, j [m], iv ES i,j A EA. Let LI n t ={A I n t : A is left-shifted in every M k,k=,...,n}. Now and Lemma 5 imply E,n,d= For any A 2 N let max 2 n d EA. A LI n BA ={A [n] :A A}. We note that if a H n and A Cn correspond under the bijection between Hn and Cn then Ba= A [n], where Ba is defined in Section.

4 C. Bey / Discrete Applied Mathematics The systems in LI n n d have the following easily verified properties: Lemma 6. Let A LI n t. Then i BA I n, t, i.e. A A 2 [n] t for all A,A 2 A, ii BA 2 [n] is an upset, i.e. B BA and B B 2 [n] imply B 2 BA. If in addition A ={A C n : A [n] BA} then iii EA = n B n B. 2 B BA and We note that iii applies for every A LI n n d with E,n,d= EA. By 2 and Lemma 6 we obtain a further reduction: E,n,d= max n B n B. 3 2 B I n,n d B B Given a family B 2 [n] and nonnegative real numbers weights ω i, i = 0,...,n, put ωb = B B ω B = n [n] B ω i i i=0 Mn, t, ω = max ωb. B I n,t We consider the weights ω i = n i n i. 4 Then, according to 3 we have E,n,d= Mn, n d,ω. 5 2 Let F i n, t ={B [n] : B [t + 2i] t + i} 2 [n], and recall the families F i,n,t H n defined in the previous section. With Lemma 6iii we obtain EF i,n,t = ωf 2 i n, t. 6 Finally, 5 and 6 show that Theorem 4 is equivalent to the following: Theorem 4. Let 3 and ω i = n i n i, i = 0,...,n. Then { ωf0 n, t if t =, 2, Mn, t, ω = max ωf in, t if t>2. 0 i n t/2 We remark that the cases t = orn sufficiently large follow from results of Frankl [7, Theorem 5.2, Remark 5.3 and Theorem 5.4].

5 54 C. Bey / Discrete Applied Mathematics Auxiliary results Lemma 7. Let S 2 [m] be a nonempty system of sets such that i S is complement-closed, i.e. A S implies A =[m]\a S and ii S is convex, i.e. A, C S and A B C imply B S. Furthermore, let ν 0 > ν > > ν m > 0 be real numbers. Put { } Q = min νi+ : 0 i<m. ν i Then there exists an intersecting subsystem I S such that νi νs. 7 + Q Proof. We follow the construction of an intersecting system I given in [4]. If S = 2 [m] we may take I ={A [m] :m A} since then + QνI = ν A + + Q ν A + ν A + + ν A = νs. A [m ] A [m ] Assume now that S = 2 [m] and choose a set B S such that B\{i} / S for some element i [m]. Consider the partition S = S S 2 S 3 S 4, where S ={A S : i A and A\{i} S}, S 2 ={A S : i/ A and A {i} S}, S 3 ={A S : i A and A\{i} / S}, S 4 ={A S : i/ A and A {i} / S}. Clearly, S ={A : A S }=S 2 and S 3 ={A : A S 3 }=S 4, and hence S = S 2 and S 3 = S 4. Further, S ={A {i} :A S 2 } and therefore νs = ν A = ν A + ν A = Q Q νs 2. 8 A S A S 2 A S 2 Now S S 3 is clearly an intersecting system. It is easily verified that also the system S S 4 is intersecting. We may assume that νs S 3 + Q νs, since otherwise we are done. Then necessarily νs 2 S 4 Q νs. 9 + Q Since B S 3 we have S 4 = S 3 =. Hence νs 4 >/QνS 4, and with 8 and 9 we obtain νs S 4 > Q νs 2 S 4 + Q νs. Remark. The proof shows that there is always an intersecting subsystem I S for which strict inequality in 7 holds unless S = 2 [m] and ν i+ /ν i is constant for i = 0,...,m.

6 A I C. Bey / Discrete Applied Mathematics Corollary 8. Let S 2 [m] be a nonempty system of sets as in Lemma 7. Further let > and c>0 be reals. Then there exists an intersecting subsystem I S with m A +c m A +c A + + /c A. 0 For m> there is an intersecting subsystem I S such that strict inequality in 0 holds. A S We continue with further auxiliary results. Recall the families F i n, t which we abbreviate here and in the next section by F i. Recall also the weights ω i which were defined in 4. Lemma 9. The sequence ωf i, i = 0,..., n t/2 is unimodal. More precisely, for 0 <r n t/2 we have ωf r ωf r if and only if + / + r + t. r Proof. The lemma follows by comparing the two numbers and ωf r \F r = t + r = t + r = t + r ωf r \F r = t + r 2 = t + r 2 n t 2r j ω j+t+r n t 2r j r n t 2r ω j j+t+r r n t 2r n t 2r We need also the following numerical fact. n t r + j n t r j + r + + r. Lemma 0. Let > and r n t/2. Then ωf r ωf r implies + + t / + s s for every positive integer s r. Moreover, for t>2 we have strict inequality in unless ωf r = ωf r and s = r. Proof. Put c = /. It suffices to show that + t / + t + / + s s c + s c + s for every integer s r. Note that the LHS is decreasing in t, the RHS is increasing in c, and equality holds for t, c = 2, 0.

7 56 C. Bey / Discrete Applied Mathematics Proof of Theorem 4 We prove Theorem 4. It follows essentially from the following two lemmas whose proofs utilize the methods from [] generating sets and [3] pushing pulling. Let LIn, t denote the set of all left-shifted t-intersecting systems B 2 [n]. Put F n t/2 + = F =. Lemma. Let r be the smallest nonnegative integer such that ωf r >ωf r+. Then every B LIn, t with ωb = Mn, t, ω is t-intersecting in [t + 2r], i.e. B B 2 [t + 2r] t for all B,B 2 B. This is a special case of [5, Lemma 29] let their,kbe our,n, and is thus not reproved here. Note that in [5] only the existence of a system B having the properties of Lemma is stated even under the weaker requirement ωf r ωf r+ ; this already would suffice for the following arguments, and that the proof there gives indeed the above stronger statement which, however, is only needed for uniqueness considerations. We remark that the existence of a system B with the properties in Lemma follows more generally for all weights which satisfy ω i ω i+, i = t,...,n, and ωf r ωf n t/2, see [6, Theorem 5]. Lemma 2. Let t 2. Let r be the largest integer such that t +2r n and ωf r <ωf r. Then every B LIn, t with ωb = Mn, t, ω is invariant under exchanging coordinates in [t + 2r], i.e. S i,j B = B for all i, j [t + 2r]. Proof. If t = 2 then r = 0 and ωf 0 >ωf by Lemma 9 note that 3. Lemma shows that B is invariant in [t]. Let t>2. We consider two cases, first let ωf r >ωf r+. By Lemma every B LIn, t with ωb=mn, t, ω is t-intersecting in [t +2r]. Let A={B [t +2r] :B B}. Then, since ωb = Mn, t, ω and ω i > 0 for i = t,...,n, necessarily and thus B ={B [n] :B [t + 2r] A}, ωb = ω A where the new weights ω t,...,ω t+2r are given by n t 2r ω i = j ω i+j = n t 2r j = t+2r n t 2r / + t + 2r i i n i j n i j Further, we clearly have A LIt + 2r, t. Let l t + 2r be the largest integer such that A is invariant under exchanging coordinates in [l]. Assume that l<t + 2r. 2 Consider the sets A ={A A : S l+,i A / A for some i l}, A ={A [l + 2,t + 2r] :A A }. Clearly, A = and hence A =. Now we state the following facts which among others follow from the pushing pulling method [3,4] see also [6]: i l t and l + t is even. ii A is complement-closed i.e. A A implies [l + 2,n]\A A, and A is convex i.e. A A 2 A 3 and A,A 3 A imply A 2 A..

8 C. Bey / Discrete Applied Mathematics iii For all intersecting subsystems I of A, ω A +l+t/2 l t + 2 ω A +l+t/2 2l +. 3 A I A A In view of i and 2 we may write l = t + 2s 2 for an integer s with s r. By ii we may apply Corollary 8 to the set system A 2 [t+2s,t+2r] ; this gives an intersecting system A A such that A A m A +c A + + /c A A m A +c A 4 for m = t + 2r l = 2r s + and any constant c. We put c = + s in order to get from 4 ω A +l+t/2 ω A +l+t/2 + + /c. 5 A A A A Now, recalling our choice of r, we obtain from Lemma /c > 2 + t /s = l t + 2 2l +, 6 which in view of 3 and 5 shows that the intersecting system A A contradicts fact iii. Thus, the assumption 2 is false, i.e. A and hence also B is invariant in [t + 2r]. Replacing r by r + in the above arguments including Lemma 0 except in 2, in the condition s r and in the conclusion that A is invariant in [t + 2r], yields a proof in the case ωf r = ωf r+. Proof of Theorem 4. Let B I n, t with Mn, t, ω = ωb. According to Lemma 5 we can assume that B LIn, t. Let r be the largest integer such that t + 2r n and ωf r ωf r hold. For t = and t = 2 we obtain from Lemma 9 note 3 that r = 0. Then Lemma gives B F 0, i.e. Mn, t, ω = ωf 0. Let t>2. By Lemma, B is t-intersecting in [t + 2r]. IfωF r <ωf r then Lemma 2 shows that B is also invariant in [t + 2r], which clearly implies B F r, i.e. Mn, t, ω = ωf r.ifωf r = ωf r then Lemma 2 shows that B is invariant in []. It follows B F r or B F r, i.e. Mn, t, ω = max{ωf r, ωf r }. References [] R. Ahlswede, L.H. Khachatrian, The complete intersection theorem for systems of finite sets, European J. Combin [2] R. Ahlswede, L.H. Khachatrian, The diametric theorem in Hamming spaces optimal anticodes, Adv. Appl. Math [3] R. Ahlswede, L.H. Khachatrian, A pushing pulling method: new proofs of intersection theorems, Combinatorica [4] R. Ahlswede, L.H. Khachatrian, A diametric theorem for edges, J. Combin. Theory Ser. A [5] Ch. Bey, The Erdős Ko Rado bound for the function lattice, Discrete Appl. Math [6] Ch. Bey, K. Engel, Old and new results for the weighted t-intersection problem via AK-methods, in: I. Althöfer, N. Cai, G. Dueck, L.H. Khachatrian, M. Pinsker, A. Sárközy, I. Wegener, Z. Zhang Eds., Numbers, Information and Complexity, Kluwer Academic Publishers, Boston, Dordrecht, London, 2000Special volume in honour of R. Ahlswede on occasion of his 60th birthday. [7] M. Deza, P. Frankl, Erdős Ko Rado theorem 22 years later, SIAM J. Algebraic Discrete Methods [8] P. Erdős, C. Ko, R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. II [9] P. Frankl, N. Tokushige, The Erdős Ko Rado theorem for integer sequences, Combinatorica [0] Gy. Katona, Intersection theorems for systems of finite sets, Acta Math. Hungar [] D. Kleitman, On a combinatorial conjecture of Erdős, J. Combin. Theory