Báo cáo toán học: "The Structure of Maximum Subsets of {1, . . . , n} with No Solutions to a + b = kc" potx

The Structure of Maximum Subsets of {1, ,n} with No Solutions to a + b = kc Andreas Baltz ∗ Mathematisches Seminar University of Kiel, D-24098 Kiel, Germany aba@numerik.uni-kiel.de Peter Hegarty, Jonas Knape, Urban Larsson Department of Mathematics Chalmers University of Technology G¨oteborg, Sweden hegarty@math.chalmers.se {md9jonas,md0larur}@mdstud.chalmers.se Tomasz Schoen ∗∗ Wydzial Matematyki i Informatyki Adam Mickiewicz University Pozna´n, Poland schoen@amu.edu.pl Submitted: Nov 9, 2004; Accepted: Apr 22, 2005; Published: Apr 28, 2005 MR Subject Classifications: 05D05, 11P99 Abstract If k is a positive integer, we say that a set A of positive integers is k-sum-free if there do not exist a, b, c in A such that a + b = kc. In particular we give a precise characterization of the structure of maximum sized k-sum-free sets in {1, ,n} for k ≥ 4andn large. 1 Introduction A set of positive integers is called k-sum-free if it does not contain elements a, b, c such that a + b = kc, ∗ supported by DFG, Grant SR 7/9 – 2 ∗∗ research partially supported by KBN Grant 2 PO3A 007 24 the electronic journal of combinatorics 12 (2005), #R19 1 where k is a positive integer. Denote by f(n, k) the maximum cardinality of a k-sum-free set in {1, ,n}.Fork = 1 these extremal sets are well-known: Deshoulliers, Freiman, S´os, and Temkin [1] proved in particular that the maximum 1–sum-free sets in {1, ,n} are precisely the set of odd numbers and the “top half” {  n+1 2  , ,n}.Forn>8even { n 2 , ,n− 1} forms the only additional extremal set. The famous theorem of Roth [4] gives f(n, 2) = o(n). Chung and Goldwasser [2] solved the case k = 3 by showing that the set of odd integers is the unique extremal set for n>22. For k ≥ 4theygaveanexample of a k-sum-free set [3] of cardinality k(k−2) k 2 −2 n + 8(k −2) k(k 2 −2)(k 4 −2k 2 −4) n + O(1), which implies lim n→∞ f(n,k) n ≥ k(k−2) k 2 −2 + 8(k −2) k(k 2 −2)(k 4 −2k 2 −4) , and they conjectured that this lower bound is the actual value. Moreover they conjectured that extremal k-sum-free sets consist of three intervals of consecutive integers with slight modifications at the end-points if n is large. In this paper we prove that the first conjecture is true, and we expose a structural result which is very close to the second. Our proof is elementary. In fact it is based on two simple observations: Suppose we are given a k-sum-free set A.Then • kx − y/∈ A for all x, y ∈ A (Otherwise we could satisfy the equation kx =(kx − y)+y in A.) • for all y ∈ A any interval centered around ky 2 cannot share more than half of its elements with A. (Otherwise we would find a pair  ky 2  − d,  ky 2  + d in A, giving  ky 2  − d  +  ky 2  + d  = ky.) 2 Preparations Let n ∈ N be large and let k ∈ N ≥4 . We start by agreeing on some notations. Notations Let A ⊆{1, ,n} be a set of positive integers. Denote by s A := min A and m A := max A the smallest and the largest elements of A respectively. For l, r ∈ R let (l, r]:={x ∈ N | l<x≤ r} [l, r):={x ∈ N | l ≤ x<r} (l, r):={x ∈ N | l<x<r} [l, r]:={x ∈ N | l ≤ x ≤ r} the electronic journal of combinatorics 12 (2005), #R19 2 abbreviate intervals of integers. Continuous intervals will be indicated by the subscript R. Furthermore for any y ∈ N and d ∈ N 0 (:= N ∪{0}) put I d y :=  ky − 1 2 − d, ky +1 2 + d  . Note that if ky is even then I d y =  ky 2 − d, ky 2 − d +1, , ky 2 + d  and |I d y | =2d +1, while if ky is odd we have I d y =  ky−1 2 − d, , ky+1 2 + d  and |I d y | =2d +2. The first Lemma restates our introductory observations. Lemma 1 Let A ⊆ [1,n] be a k-sum-free set. If x, y ∈ A then kx − y/∈ A.Ify ∈ A and d ∈ N 0 then |I d y \ A|≥d +1. Suppose A  is a k-sum-free set consisting of intervals (l i ,r i ]. The interval (l i ,r i ]isk- sum-free if l i ≥ 2r i k . Moreover we observe that reasonably large consecutive intervals (l i+1 ,r i+1 ], (l i ,r i ] (where we assume r i+1 <l i ) should satisfy kr i+1 ≤ l i +s A  . This leads to the following definition, describing a successive transformation of an arbitrary k-sum-free set A into a k-sum-free set of intervals. Definition 1 Let n ∈ N and let A ⊆ [1,n] be k-sum-free with smallest element s := s A . Define sequences (r i ), (l i ), (A i ) by: A 0 := A, r 1 := n, l i :=  2r i k  ,r i+1 :=  l i + s k  , A i := (A i−1 \ (r i+1 ,l i ]) ∪ (l i ,r i ] ∩ [s, n] for i ≥ 1. The letter t = t A will be reserved to denote the least integer such that r t+1 <s. Observe that, for all i ≥ t, A i = A t =[α, r t ] ∪  t−1  j=1 (l j ,r j ]  , (1) where α = α A := max{l t +1,s}. 3 The structure of maximum k-sum-free sets To obtain the structural result we consider the successive transformation of an arbitrary k-sum-free set A intoasetA t of intervals as in (1). Our plan is to show that each member of the transformation sequence (A i )isk-sum-free and has size greater than or equal to |A|.Forn sufficiently large, depending on k, and a maximum sized k-sum-free subset A of [1,n], it will turn out that A t consists of three intervals only, i.e.: that t =3. This observation will do to determine f(n, k), and we conclude our proof by showing that A the electronic journal of combinatorics 12 (2005), #R19 3 could be enlarged if it did not contain (nearly) the whole interval (l 3 ,r 3 ] and consequently almost all elements from (l 2 ,r 2 ]and(l 1 ,r 1 ], so that in fact almost nothing happens during the transformation of an extremal set. Lemma 2 Let A ⊆ [1,n] be k-sum-free. Let i ∈ N. a) A i is k-sum-free. b) |A i |≥|A i−1 |. Proof. a) Clearly, it is enough to prove the claim for i ≤ t, so we may assume that s ≤ r i . Suppose there are a, b, c ∈ A i with a + b = kc. A i is of the form A i = A i−1 ∩ [s, r i+1 ] ∪ (l i ,r i ] ∩ [s, n] ∪ (l i−1 ,r i−1 ] ∪ ∪ (l 1 ,r 1 ]. If c ∈ (l 1 ,r 1 ], then kc > 2n, which is impossible. If i ≥ 2andc ∈ (l j ,r j ] for some j ∈ [2,i], then kc ∈ (2r j ,l j−1 + s] and the larger one of a, b must be in (r j ,l j−1 ]. But (r j ,l j−1 ] ∩ A i = ∅ by construction. Hence c ∈ A i−1 ∩ [s, r i+1 ]. Now, kc ≤ kr i+1 ≤ l i + s. Since (r i+1 ,l i ] ∩ A i = ∅,botha and b have to be in A i−1 ∩ [s, r i+1 ]=A ∩ [s, r i+1 ]. But A is k-sum-free, a contradiction. b) The inequality is trivial for i ≥ t.For1≤ i<twe have that l i ≥ s and hence A i =(A i−1 ∩ [1,r i+1 ]) ∪ (l i ,r i ] ∪  i−1  j=1 (l j ,r j ]  . Thus it suffices to prove that |A i−1 ∩ [1,r i ]|≤|A i−1 ∩ [1,r i+1 ]| +  (k − 2)r i k  . Clearly, then, it suffices to prove the inequality for i = 1, i.e.: to prove that, for any n>0, and any k-sum-free subset A of [1,n] with smallest element s A ,wehave |A|≤|A ∩ [1,r 2,A ]| +  (k − 2)n k  , (2) where r 2,A :=  2n/k + s A k  . The proof is by induction on n. The result is trivial for n = 1. So suppose it holds for all 1 ≤ m<nand let A be a k-sum-free subset of [1,n]. Note that the result is again trivial if s A > 2n/k, so we may assume that s A ≤ 2n/k, which implies that r 2,A ≤ n/k,sincek ≥ 4. First suppose that there exists x ∈ A ∩ (n/k, 2n/k]. Then 1 ≤ kx − n ≤ n and the the electronic journal of combinatorics 12 (2005), #R19 4 map f : y → kx − y is a 1-1 mapping from the interval [kx − n, n] to itself. For each y in this interval, at most one of the numbers y and f(y) can lie in A,sinceA is k-sum-free. To simplify notation, put w := kx − n − 1. Then our conclusion is that |A ∩ (w, n]|≤ 1 2 (n − w). (3) If w =0orifA ∩ [1,w]=∅ , then we are done (since k ≥ 4). Put B := A ∩ [1,w]. Then we may assume B = ∅, hence s B = s A . Applying the induction hypothesis to B, we find that |B| = |A ∩ [1,w]|≤|B ∩ [1,r 2,B ]| +  (k − 2)w k  . (4) But s B = s A implies that r 2,B ≤ r 2,A , hence that B ∩ [1,r 2,B ] ⊆ A ∩ [1,r 2,A ]. Thus (3) and (4) yield the inequality |A|≤|A ∩ [1,r 2,A ]| +  (k − 2)w k  + 1 2 (n − w), which in turn implies (2), since |A| is an integer. Thus we are reduced to completing the induction under the assumption that A ∩ (n/k, 2n/k]=∅. Suppose x ∈ A ∩ (r 2,A ,n/k]. Then 2n/k + s A <kx≤ n and kx − s A ∈ A. In other words, we can pair off elements in A ∩ (r 2,A , 2n/k] with elements in (2n/k, n]\A. This immediately implies (2), and the proofofLemma2iscomplete.  We have seen so far that any k-sum-free set A can be turned into a k-sum-free set A t having overall size at least |A|.ThesetA t is a union of intervals, as given by (1), though note that the final interval [α, r t ] may consist of a single point, since r t = s is possible. The proof of the following Lemma uses a fact shown in [3] by Chung and Goldwasser, to prove that t must be equal to three if |A| is maximum. Lemma 3 Let A be a maximum k-sum-free subset of [1,n], where n>n 0 (k) is sufficiently large. Let s := s A and let t := max{i ∈ N | r i ≥ s}. Then t =3. Proof. Let A t be the set of positive integers given by (1). In a similar manner we now define a k-sum-free subset A  t of (0, 1] . Put c := s/n and, for i =1, , t define real numbers R i ,L i as follows : R 1 := 1,L i := 2R i k ,R i+1 := L i + c k . Then we put A  t := [α  ,R t ) ∪  t−1  j=1 [L j ,R j )  , the electronic journal of combinatorics 12 (2005), #R19 5 where α  := max{L t ,c}.ThatA  t is k-sum-free is shown in [3]. One sees easily that |A t |≤n · µ(A  t )+t, (5) where µ denotes the Lebesgue-measure. Now suppose that t = 3. It is shown in [3] that there exists a constant c k > 0, depending only on k, such that in this case |µ(A  t )|≤ k(k − 2) k 2 − 2 + 8(k − 2) k(k 2 − 2)(k 4 − 2k 2 − 4) − c k . (6) In fact, in the notation of page 8 of [3], an explicit value for c k (which we will use later) is given by c k = 2 k (R(3) − R(4)), which by definition of R amounts to c k = 8(k 4 − 4k 2 − 4)(k − 2) (k 6 − 2k 4 − 4k 2 − 8)(k 4 − 2k 2 − 4)k . (7) Now (5) and (6) would imply that |A|≤ k(k − 2) k 2 − 2 n + 8(k − 2) k(k 2 − 2)(k 4 − 2k 2 − 4) n − c k n + t. But we have seen in the introduction that |A|≥ k(k−2) k 2 −2 n + 8(k −2) k(k 2 −2)(k 4 −2k 2 −4) n + O(1) and, since t = O(log k n), we thus have a contradiction for sufficiently large n. Hence t must equal three, for large enough n, as required.  Now we are nearly in a position to determine f(n, k). We want to calculate the car- dinality of an extremal k-sum-free set A via computing |A 3 |.Since|A 3 | depends on s A , the following lemma will be helpful : Lemma 4 Let n>n 0 (k) be sufficiently large. If A isamaximalk-sum-free subset of [1,n], then S − 2k ≤ s A ≤ S +3, where S :=  8n k 5 −2k 3 −4k . Proof. Set s := s A . By Lemma 3, for n>n 0 (k)wehaver 4 <s.SinceA is maximal we have |A| = |A 3 |. Now, for a fixed n, the cardinality of A 3 is a function of s ∈ [1,n] only. So we need to show that |A 3 (s)| attains its maximum value only for some s ∈ [S − 2k,S +3]. Define s  := min{s ∈ [1,n]:l 3 (s) <s}. A tedious computation (see the Appendix below) yields that s  = S +1ifk is even and s  = S or S +1ifk is odd. Hence s  ∈ [S, S +1]. (8) the electronic journal of combinatorics 12 (2005), #R19 6 Clearly, |A 3 (s)| =   (k−2)n k  + r 2 (s) − l 2 (s)+r 3 (s) − s +1, if s ≥ s  ,  (k−2)n k  + r 2 (s) − l 2 (s)+r 3 (s) − l 3 (s), if s<s  . (9) How does | A 3 (s)| change (ignoring its maximality for a while) if we alter s? First suppose s ≥ s  .Ifs increases by one, then |A 3 | will decrease by one unless either r 2 or r 3 increases. Now r 2 can only increase (by one) once in k(≥ 4) times. Almost the same is true of r 3 , though its dependence on l 2 makes things a little more complicated. However, it is not hard to see that we encounter an irreversible decrease in the cardinality of |A 3 | after at most 3 steps of increment of s. Hence |A 3 (s)| < |A 3 (s  )| if s ≥ s  +3. Next suppose s<s  . If we decrease s,then|A 3 | cannot increase at all, since l i will not decrease unless r i does. Moreover, |A 3 | will become smaller if the size of any interval is di- minished. So we can focus our attention on (l 2 ,r 2 ]. While r 2 decreases once in k times, l 2 does so no more than once in k k 2 ≥2k times. Thus |A 3 (s)| < |A 3 (s  −1)| if s ≤ s  −1−2k. We have now shown that, as a function of s ∈ [1,n], the cardinality of A 3 attains its maximum only for some s ∈ [s  − 2k, s  + 2]. This, together with (8), completes the proof of the lemma.  Now we can prove the first conjecture of Chung and Goldwasser. Theorem 1 lim n→∞ f(n, k) n = k(k − 2) k 2 − 2 + 8(k − 2) k(k 2 − 2)(k 4 − 2k 2 − 4) . Proof. Let A be a maximum k-sum-free set in [1,n], with n sufficiently large. From Lemma 4 we have s A n = S ∗ n + o(1), where S ∗ = 8n k 5 −2k 3 −4k . Thus we can estimate f(n, k) n = |A 3 | n = r 1 − l 1 + r 2 − l 2 + r 3 − S ∗ +1 n + o(1) = 1 n  n − 2n k + 2n + kS ∗ k 2 − 4n +2kS ∗ k 3 + 4n +2kS ∗ + k 3 S ∗ k 4 − S ∗  + o(1) = k 4 − 2k 3 +2k 2 − 4k +4 k 4 + S ∗ nk 3 (2k 2 − 2k +2− k 3 )+o(1) = k 4 − 2k 3 +2k 2 − 4k +4 k 4 + 8(2k 2 − 2k +2− k 3 ) (k 5 − 2k 3 − 4k)k 3 + o(1) = k 5 − 2k 4 − 4k +8 (k 4 − 2k 2 − 4)k + o(1) = k(k − 2) k 2 − 2 + 8(k − 2) k(k 2 − 2)(k 4 − 2k 2 − 4) + o(1), and the claim follows by taking the limit.  We can now show the main result. the electronic journal of combinatorics 12 (2005), #R19 7 Theorem 2 Let k ∈ N ≥4 and n>n 1 (k).LetS and s  be as in Lemma 4. Let A ⊆ {1, ,n} be a k-sum-free set of maximum cardinality, with smallest element s = s A . Then s ∈ [S, S +3]and A = I 3 ∪I 2 ∪I 1 , where I 3 ∈  {[s, r 3 ], [s, r 3 +1]} , if s ≥ s  {[s, r 3 ), [s, r 3 ] \{r 3 − 1}}, if s<s  , I 2 ∈  {[l 2 +2,r 2 ], [l 2 +2,r 2 +1]} , if r 3 +1∈ A {(l 2 ,r 2 ], (l 2 ,r 2 +1], [l 2 ,r 2 ), [l 2 ,r 2 ] \{r 2 − 1}}, if r 3 +1 /∈ A, I 1 ∈  {[l 1 +2,n]}, if r 2 +1∈ A {[l 1 ,n), (l 1 ,n], [l 1 ,n] \{n − 1}}, if r 2 +1 /∈ A, If k is even, then I i =[l i ,r i ] \{r i − 1} for 1 ≤ i ≤ 3. Remark. Note that Theorem 2 does not precisely determine the k-sum-free subsets of {1, , n} of maximum size, for every n>n 1 (k). With n and k fixed, one first needs to determine for which value(s) of s ∈ [S, S +3] the quantity |A 3 (s)|, as given by (9), is maximized. The result will depend on n and k. Even then, for a fixed s, not all the possibilities for I 3 ∪I 2 ∪I 1 need be k-sum-free. See Section 4 below for further discussion. Proof. We have already seen that |A 3 | = |A|. Our first aim is to show by compar- ing A 3 with A 2 that almost the whole interval (l 3 ,r 3 ]mustbeinA. Having achieved this, we infer by Lemma 1 that (r 3 ,l 2 ] ∩ A is nearly empty. Comparing A 2 with A 1 will then reveal that most of (l 2 ,r 2 ] is contained in A. Again Lemma 1 will help us to see that A cannot share many elements with (r 2 ,l 1 ] and a final comparison of A 1 with A will conclude the proof. (I) The first aim is easily reached if s := s A ≥ l 3 + 1. Simply note that A 2 =(A ∩ [s, r 3 ]) ∪ (l 2 ,r 2 ] ∪ (l 1 ,r 1 ] ⊆ [s, r 3 ] ∪ (l 2 ,r 2 ] ∪ (l 1 ,r 1 ]=A 3 . The maximality of |A 2 | gives A 2 = A 3 and hence [s, r 3 ] ⊆ A. Observe that s>l 3 together with Lemma 4 and (8) give S ≤ s ≤ S +3. Assume now that s ≤ l 3 . We want to show that in this case s = l 3 . Suppose s<l 3 and let B =[S − 2k, l 3 ] ∩ A. Define C := I 1 s B ∪  b∈B\{s B } I 0 b . Clearly C ⊆ (l 3 ,r 3 ] for all n  0. Then since C is the union of disjoint intervals, Lemma 1givesthat|C \ A| > |B|. Hence we get the contradiction |A 3 | = |(A 2 \ B) ∪ (l 3 ,r 3 ]|≥ |(A 2 \ B) ∪ (C \ A)| > |A 2 |−|B| + |B| = |A 2 |. Therefore we are left with s = l 3 ,andthis implies |A 2 | = |A 3 |⇐⇒|A ∩ [s, r 3 ]| = |(l 3 ,r 3 ] ∩ [s, r 3 ]| = |(s, r 3 ]|. (10) the electronic journal of combinatorics 12 (2005), #R19 8 If r 3 /∈ A we can infer from (10) that A ∩ [s, r 3 ]=[s, r 3 − 1] = [l 3 ,r 3 − 1]. If r 3 ∈ A, Lemma 1 gives kl 3 − r 3 /∈ A,so−k +1≤ kl 3 − 2r 3 ≤−1. If kl 3 − 2r 3 ≤−2we get I 1 l 3 ⊆ (l 3 ,r 3 ]and|I 1 l 3 \ A|≥2, which is impossible since this would imply |A 3 | > |A 2 |. Hence kl 3 − 2r 3 = −1andk is odd. Using (10) one obtains A ∩ [s, r 3 ]=[l 3 ,r 3 ] \{r 3 − 1}. Suppose now that s = l 3 and r 3 +1∈ A.Thenkl 3 − (r 3 +1)∈ A and r 3 − k ≤ kl 3 − (r 3 +1)≤ r 3 − 1. This contradicts that [s, r 3 − 2] ⊆ A unless kl 3 − (r 3 +1)=r 3 − 1, but then r 3 ∈ A and |A ∩ [s, r 3 ]| = |A ∩ [s, r 3 − 2]| which contradicts (10). Hence r 3 +1 /∈ A if s = l 3 . Finally note that, if s = l 3 and kl 3 ≥ 2r 3 − 1, the latter being a requirement for ei- ther of the two possibilities for I 3 to be k-sum-free, then another computation similar to the one in the Appendix yields that s ≥ S. Again, using Lemma 4 we obtain S ≤ s ≤ S +3, (11) as claimed in the statement of the theorem. This completes the first part of our proof. (II) For the second part note that we have just shown s ≥ l 3 . (12) Plugging (11) into the definition of l 3 yields (after a further tedious computation similar to that in the Appendix) S − 1 ≤ l 3 ≤ S +1, (13) which implies in view of (12) and (11) l 3 ≤ s ≤ l 3 +4. (14) Moreover we have observed that [s, r 3 −2] ⊆ A. Let ξ 1 , ,ξ 5 ∈{0, ,k−1} be constants such that kl 1 =2r 1 − ξ 1 (15) kr 2 = l 1 + s − ξ 2 (16) kl 2 =2r 2 − ξ 3 (17) kr 3 = l 2 + s − ξ 4 (18) kl 3 =2r 3 − ξ 5 . (19) the electronic journal of combinatorics 12 (2005), #R19 9 We suppose that n is sufficiently large, so we can be sure that [ks − (r 3 − 2),k(r 3 − 2) − s] ∩ A = ∅. By (14) we can infer that ∅ =[k(l 3 +4)− (r 3 − 2),k(r 3 − 2) − s] ∩ A =[r 3 − ξ 5 +4k +2,l 2 − ξ 4 − 2k] ∩ A. Let J =[r 3 +2,r 3 − ξ 5 +4k +1]∩ A and K =  x∈J {kx − (s +2),kx− (s +1),kx− s}. Then K ∩ A = ∅, |K| =3|J| and by (18) and (19) we have K ⊆ [l 2 − ξ 4 +2k − 2,l 2 − ξ 4 − kξ 5 +4k 2 + k] ⊆ (l 2 + k − 2,l 2 +4k 2 + k] ⊆ (l 2 +2,r 2 ], if n  0. Let B =[l 2 − ξ 4 − 2k +1,l 2 ] ∩ A.IfB ∪ J ⊆{l 2 } then A ∩ [r 3 +2,l 2 − 1] = ∅. Otherwise, with C as in part (I) if |B| > 1 we can verify that C ⊆ [r 2 − 3k 2 −k+2 2 ,r 2 ] ⊆ (l 2 +1,r 2 ], for n  0, and |C \ A| > |B|.PutC := ∅ if |B|≤1. For large n, K and C are disjoint. Hence |B ∪ J| < |(C \ A) ∪ K| and we get |A 2 | = | [A 1 \ (J ∪ B ∪{r 3 +1})] ∪ (l 2 ,r 2 ]| > |A 1 \{r 3 +1}|. Thus if r 3 +1∈ A we get |A 2 | > |A 1 | so suppose r 3 +1∈ A. Then neither l 2 nor l 2 +1 can be in A 1 . Otherwise, since (s − ξ 4 + k),s− ξ 4 + k − 1 ∈ [s, s + k] ⊆ [s, r 3 − 2] ⊆ A we get k(r 3 +1)=l 2 +(s − ξ 4 + k)=(l 2 +1)+(s − ξ 4 + k − 1), which is impossible. But l 2 +1∈ A 2 , so also in this case it follows that |A 2 | > |A 1 |,since l 2 +1∈ K ∪ C for large n. Again we conclude that A ∩ [r 3 +2,l 2 − 1] = ∅. Consequently, |A 2 | = |A 1 |⇔|A ∩ ([l 2 ,r 2 ] ∪{r 3 +1})| = |(l 2 ,r 2 ]|, which gives A∩ [l 2 ,r 2 ]=[l 2 +2,r 2 ]ifr 3 +1 ∈ A. If r 3 +1 /∈ A and either l 2 /∈ A or r 2 /∈ A, we get A ∩ [l 2 ,r 2 ]=(l 2 ,r 2 ]orA ∩ [l 2 ,r 2 ]=[l 2 ,r 2 ), respectively. In case r 3 +1 /∈ A and both l 2 ,r 2 ∈ A, we see that kl 2 − r 2 = r 2 − ξ 3 /∈ A.Ifξ 3 ≥ 2thenI 1 l 2 ⊆ (l 2 ,r 2 ]andl 2 could be profitably replaced. Hence ξ 3 =1,A ∩ [l 2 ,r 2 ]=[l 2 ,r 2 ] \{r 2 − 1} and k is odd. (III) For the final interval (l 1 ,r 1 ] we use Lemma 1 to conclude from [s, r 3 − 2] ⊆ A and [l 2 +2,r 2 − 2] ⊆ A in view of (16) and (17) that, for n  0, ∅ = A ∩ [k(l 2 +2)− (r 2 − 2),k(r 2 − 2) − (l 2 +2)] = A ∩ [r 2 − ξ 3 +2k +2,l 1 + s − ξ 2 − 2k − l 2 − 2], and ∅ = A ∩ [k(l 2 +2)− (r 3 − 2),k(r 2 − 2) − s] = A ∩ [2r 2 − ξ 3 +2k − r 3 +2,l 1 − ξ 2 − 2k] the electronic journal of combinatorics 12 (2005), #R19 10 [...] .. .< /b> n ], < /b> [s, r3 ] ∪ [l2 + < /b> 1, < /b> r2 + < /b> 1] ∪ [l1 + < /b> 2, < /b> n ], < /b> [s, r3 + < /b> 1] ∪ [l2 + < /b> 2, < /b> r2] ∪ [l1 , < /b> n − 1 ], < /b> [s, r3 + < /b> 1] ∪ [l2 + < /b> 2, < /b> r2] ∪ [l1 + < /b> 1, < /b> n ], < /b> [s, r3 + < /b> 1] ∪ [l2 + < /b> 2, < /b> r2 + < /b> 1] ∪ [l1 + < /b> 2, < /b> n] Note that, by Theorem 2, < /b> for a < /b> given n ≥ 1100 8, < /b> every maximal 4-sum-free subset of < /b> { 1, < /b> < /b> , < /b> n} < /b> is one of < /b> the sets Jx (s ), < /b> for some s ∈ [S, S + < /b> 3] = [s − 1, < /b> s + < /b> 2] By the remarks above, for each i ∈ { 0, < /b> , 109 }, < /b> there are natural .. .< /b> = = = = = = = = = = = = = [S, r3 − 1] ∪ [l2 , < /b> r2 − 1] ∪ [l1 , < /b> n − 1 ], < /b> [S, r3 − 1] ∪ [l2 , < /b> r2 − 1] ∪ [l1 + < /b> 1, < /b> n ], < /b> [S, r3 − 1] ∪ [l2 + < /b> 1, < /b> r2] ∪ [l1 , < /b> n − 1 ], < /b> [S, r3 − 1] ∪ [l2 + < /b> 1, < /b> r2] ∪ [l1 + < /b> 1, < /b> n ], < /b> [S, r3 − 1] ∪ [l2 + < /b> 1, < /b> r2 + < /b> 1] ∪ [l1 + < /b> 2, < /b> n ], < /b> [s, r3 ] ∪ [l2 , < /b> r2 − 1] ∪ [l1 , < /b> n − 1 ], < /b> [s, r3 ] ∪ [l2 , < /b> r2 − 1] ∪ [l1 + < /b> 1, < /b> n ], < /b> [s, r3 ] ∪ [l2 + < /b> 1, < /b> r2 ] ∪ [l1 , < /b> n − 1 ], < /b> [s, r3 ] ∪ [l2 + < /b> 1, < /b> r2 ] ∪ [l1 + < /b> 1 ,. . . 101 and Fi = {J9 (s + < /b> 1)} If |Fi| = 2, < /b> then Fi is {J9 (s ), < /b> J9 (s + < /b> 1)} if i = 9 3, < /b> 10 3, < /b> 10 5, < /b> 10 7, < /b> {J4 , < /b> J9 (s )} if i = 9, < /b> 1 1, < /b> 1 3, < /b> 2 5, < /b> 2 7, < /b> {J8 (s ), < /b> J9 (s )} if i = 4 8, < /b> 5 0, < /b> 5 6, < /b> 5 8, < /b> 6 0, < /b> 7 2, < /b> 7 4, < /b> 7 6, < /b> 8 8, < /b> 90 {J7 (s ), < /b> J9 (s )} if i = 6 3, < /b> 6 5, < /b> 6 7, < /b> 7 9, < /b> 81 If |Fi| = 3: F8 = F24 F15 F29 F39 F62 = F78 F53 F83 F92 F95 = F97 F102 F109 = = = = = = = = = = = {J4 , < /b> J8 (s ), < /b> J9 (s ) }, < /b> {J4 , < /b> J7 (s ), < /b> J9 (s ) }, < /b> {J4 ,. . . ), < /b> J9 (s + < /b> 1) }, < /b> {J9 (s ), < /b> J12 (s ), < /b> J9 (s + < /b> 1) }, < /b> {J6 (s ), < /b> J7 (s ), < /b> J9 (s ) }, < /b> {J9 (s ), < /b> J10 (s ), < /b> J9 (s + < /b> 1) }, < /b> {J7 (s ), < /b> J9 (s ), < /b> J9 (s + < /b> 2) }, < /b> {J8 (s ), < /b> J9 (s ), < /b> J9 (s + < /b> 1) }, < /b> {J7 (s ), < /b> J9 (s ), < /b> J9 (s + < /b> 1) }, < /b> {J9 (s ), < /b> J8 (s + < /b> 1 ), < /b> J9(s + < /b> 1) }, < /b> {J9 (s ), < /b> J7 (s + < /b> 1 ), < /b> J9(s + < /b> 1)} F1 = F3 = F17 F10 = F12 = F26 F38 F41 = F43 F52 = F64 = F66 = F80 F104 = F106 F69 = = = = = = = = {J2 , < /b> J4 , < /b> J7 (s ), < /b> J9 (s )} ,. . . = = = = = {J3 , < /b> J4 , < /b> J6 (s ), < /b> J7 (s ), < /b> J9 (s ) }, < /b> {J2 , < /b> J4 , < /b> J7 (s ), < /b> J9 (s ), < /b> J9 (s + < /b> 2) }, < /b> {J3 , < /b> J4 , < /b> J8 (s ), < /b> J9 (s ), < /b> J9 (s + < /b> 1) }, < /b> {J4 , < /b> J7 (s ), < /b> J9 (s ), < /b> J12 (s ), < /b> J9 (s + < /b> 1) }, < /b> {J6 (s ), < /b> J7 (s ), < /b> J8 (s ), < /b> J9 (s ), < /b> J9 (s + < /b> 2) }, < /b> {J6 (s ), < /b> J7 (s ), < /b> J9 (s ), < /b> J8 (s + < /b> 1 ), < /b> J9(s + < /b> 1) }, < /b> {J7 (s ), < /b> J9 (s ), < /b> J9 (s + < /b> 1 ), < /b> J10 (s + < /b> 1 ), < /b> J9(s + < /b> 2) }, < /b> {J8 (s ), < /b> J9 (s ), < /b> J6 (s + < /b> 1 ), < /b> J7 (s + < /b> 1 ), < /b> J9 (s + < /b> 1)} If |Fi| =. . . F96 = = = = = = {J2 , < /b> J4 , < /b> J7 (s ), < /b> J9 (s ), < /b> J10 (s ), < /b> J9 (s + < /b> 1) }, < /b> {J2 , < /b> J4 , < /b> J7 (s ), < /b> J9 (s ), < /b> J12 (s ), < /b> J9 (s + < /b> 1) }, < /b> {J4 , < /b> J9 (s ), < /b> J12 (s ), < /b> J13 (s ), < /b> J7 (s + < /b> 1 ), < /b> J9 (s + < /b> 1) }, < /b> {J6 (s ), < /b> J7 (s ), < /b> J8 (s ), < /b> J9 (s ), < /b> J10 (s ), < /b> J9 (s + < /b> 1) }, < /b> {J7 (s ), < /b> J9 (s ), < /b> J10 (s ), < /b> J9 (s + < /b> 1 ), < /b> J12 (s + < /b> 1 ), < /b> J9 (s + < /b> 2) }, < /b> {J6 (s ), < /b> J7 (s ), < /b> J8 (s ), < /b> J9 (s ), < /b> J8 (s + < /b> 1 ), < /b> J9 (s + < /b> 1)} If |Fi| = 7: F0 = F16 = {J1 , < /b> J2 ,. . . J4 , < /b> J6 (s ), < /b> J7 (s ), < /b> J8 (s ), < /b> J9 (s ) }, < /b> F40 = {J4 , < /b> J8 (s ), < /b> J9 (s ), < /b> J11 (s ), < /b> J12 (s ), < /b> J8 (s + < /b> 1 ), < /b> J9 (s + < /b> 1)} If |Fi| = 8: F2 F21 F30 F35 F42 F98 = = = = = = {J1 , < /b> J2 , < /b> J3 , < /b> J4 , < /b> J6 (s ), < /b> J7 (s ), < /b> J8 (s ), < /b> J9 (s ) }, < /b> {J2 , < /b> J4 , < /b> J7 (s ), < /b> J9 (s ), < /b> J10 (s ), < /b> J9 (s + < /b> 1 ), < /b> J12 (s + < /b> 1 ), < /b> J9 (s + < /b> 2) }, < /b> {J3 , < /b> J4 , < /b> J6 (s ), < /b> J7 (s ), < /b> J9 (s ), < /b> J12 (s ), < /b> J8 (s + < /b> 1 ), < /b> J9 (s + < /b> 1) }, < /b> {J2 , < /b> J4 , < /b> J7 (s ), < /b> J9 (s ), < /b> J1 2.. < /b> . (s ), < /b> J9 (s + < /b> 1 ), < /b> J10 (s + < /b> 1 ), < /b> J9 (s + < /b> 2) }, < /b> {J3 , < /b> J4 , < /b> J8 (s ), < /b> J9 (s ), < /b> J11 (s ), < /b> J12 (s ), < /b> J8 (s + < /b> 1 ), < /b> J9 (s + < /b> 1) }, < /b> {J6 (s ), < /b> J7 (s ), < /b> J8 (s ), < /b> J9 (s ), < /b> J8 (s + < /b> 1 ), < /b> J9(s + < /b> 1 ), < /b> J10 (s + < /b> 1 ), < /b> J9 (s + < /b> 2)} If |Fi| = 9: F18 = {J1 , < /b> J2 , < /b> J3 , < /b> J4 , < /b> J6 (s ), < /b> J7 (s ), < /b> J8 (s ), < /b> J9 (s ), < /b> J9 (s + < /b> 2) }, < /b> F84 = {J6 (s ), < /b> J7 (s ), < /b> J8 (s ), < /b> J9 (s ), < /b> J10 (s ), < /b> J9 (s + < /b> 1 ), < /b> J12 (s + < /b> 1 ), < /b> J8(s + < /b> 2 ), < /b> J9 (s + < /b> 2)} If |Fi| =. . . (s + < /b> 1) }, < /b> F20 = {J1 , < /b> J2 , < /b> J3 , < /b> J4 , < /b> J6 (s ), < /b> J7 (s ), < /b> J8 (s ), < /b> J9 (s ), < /b> J10 (s ), < /b> J9 (s + < /b> 1 ), < /b> J12 (s + < /b> 1 ), < /b> J8 (s + < /b> 2 ), < /b> J9 (s + < /b> 2) }, < /b> F34 = {J1 , < /b> J2 , < /b> J3 , < /b> J4 , < /b> J6 (s ), < /b> J7 (s ), < /b> J8 (s ), < /b> J9 (s ), < /b> J11 (s ), < /b> J12 (s ), < /b> J8 (s + < /b> 1 ), < /b> J9(s + < /b> 1 ), < /b> J10 (s + < /b> 1 ), < /b> J9 (s + < /b> 2)} Note, in particular, that |F ( 4, < /b> n)| ≤ 14 for all sufficiently large n Computer simulations suggest the same may be true for any even k, with < /b> .. .< /b> correspondences between the sets in the families F ( 4, < /b> n) for all n ≡ i (mod 110) By slight abuse of < /b> notation, we denote any such family simply by Fi Our computer program yielded the following result : If |Fi | = 1, < /b> then i = 6, < /b> 7, < /b> 2 2, < /b> 2 3, < /b> 4 6, < /b> 4 7, < /b> 4 9, < /b> 5 1, < /b> 5 4, < /b> 5 5, < /b> 5 7, < /b> 5 9, < /b> 6 1, < /b> 7 0, < /b> 7 1, < /b> 7 3, < /b> 7 5, < /b> 7 7, < /b> 8 6, < /b> 8 7, < /b> 89 the electronic journal of < /b> combinatorics 12 (2005 ), < /b> #R19 12 or 91 and Fi = {J9 (s ) }, < /b> or i = 3 6, < /b> 3 7, < /b> 100 or . Structure of Maximum Subsets of { 1, ,n} with No Solutions to a + b = kc Andreas Baltz ∗ Mathematisches Seminar University of Kiel, D-24098 Kiel, Germany aba@numerik.uni-kiel.de Peter Hegarty,. be large and let k ∈ N ≥4 . We start by agreeing on some notations. Notations Let A { 1, ,n} be a set of positive integers. Denote by s A := min A and m A := max A the smallest and the largest. (r i+1 ,l i ] ∩ A i = ∅,botha and b have to be in A i−1 ∩ [s, r i+1 ] =A ∩ [s, r i+1 ]. But A is k-sum-free, a contradiction. b) The inequality is trivial for i ≥ t.For1≤ i<twe have that l i ≥