The Anti-Ramsey Problem for the Sidon equation

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1 The Anti-Ramsey Problem for the Sidon equation Vladislav Taranchuk Craig Timmons August 0, 018 arxiv: v1 [math.co] 9 Aug 018 Abstract For n k 4, let ARX+Y=Z+T k (n be the maximum number of rainbow solutions to the Sidon equation X + Y = Z + T over all k-colorings c : [n] [k]. It can be shown that the total number of solutions in [n] to the Sidon equation is n /1 + O(n and so, trivially, ARX+Y=Z+T k (n n /1 + O(n. We improve this upper bound to ( 1 ARX+Y=Z+T k (n 1 1 n +O k (n 4k for all n k 4. Furthermore, we give an explicit k-coloring of [n] with more rainbow solutions to the Sidon equation than a random k-coloring, and gives a lower bound of ( n O k (n AR k k X+Y=Z+T(n. When k = 4, we use a different approach based on additive energy to obtain an upper bound of n /96+O(n, whereas our lower bound is n /96 O(n in this case. 1 Introduction Most of the notation we use is standard. For a positive integer n, let [n] = {1,,...,n}. If X is a set and m 0 is an integer, then ( X m is the set of all subsets of X of size m. A k-coloring of a set X is a function c : X [k]. The function c need not be onto. A subset Y X is monochromatic under c if c(y = c(y for all y,y Y. The set Y is rainbow if no two elements of Y have been assigned the same color. The hypergraph Ramsey Theorem states that for any positive integers s, k, and m, there is an N = N(s,k,m such that for all n N the following holds: if c is any k-coloring of ( ( [n] m, then there is a set S [n] such that S m is monochromatic under c. This theorem is one of the most important theorems in combinatorics. Today, Ramsey Theory is a cornerstone in combinatorics and there is a vast amount of literature on Ramsey type problems. Here we will focus on a Ramsey problem in the integers and Department of Mathematics and Statistics, California State University Sacramento. Department of Mathematics and Statistics, California State University Sacramento. This work was supported by a grant from the Simons Foundation (#59419, Craig Timmons. 1

2 recommend Landman and Robertson [4] for a more comprehensive introduction to this area. The problem we consider is inspired by the investigations of two recent papers. In [6], Saad and Wolf introduced an arithmetic analog of some problems in graph Ramsey Theory. In particular, given a graph H, let RM k (H,n be the minimum number of monochromatic copies of H over all k-colorings c : E(K n [k]. The parameter RM k (H,n is the Ramsey multiplicity of H and has been studied for different graphs H. The arithmetic analog from[6] replaces graphs with linear equations, and sets up a general framework where these ideas from graph theory(sidorenko property, Ramsey multiplicty have natural counterparts. Fix an abelian group Γ. If L is a linear equation with integer coefficients, one can look at the minimum number of monochromatic solutions to L over all k-colorings c : Γ [k]. One of the first examples given in [6] (see Example 1.1 concerns the Sidon equation X +Y = Z+T. This famous equation has a rich history in combinatorics. A Sidon set in an abelian group Γ is a set having only trivial solutions to X + Y = Z + T. If A Γ is a Sidon set and Γ is finite, then a simple counting argument gives A Γ 1/ + 1. The constant can be improved in many cases, but what concerns us here is that when A is much larger, say A = α Γ for some α > 0, then A will certainly contain nontrivial solutions to the Sidon equation. Thus, a natural question is given a k-coloring c : Γ [k], at least how many solutions to the Sidon equation must be monochromatic. This question, and several others including results on X +Y = Z (Schur triples and X +Y = Z ( term a.p. s, is answered by the results of [6]. For more in this direction, we refer the reader to that paper. Recently, De Silva, Si, Tait, Tunçbilek, Yang, and Young [1] studied a rainbow version of Ramsey multiplicity. Instead of looking at the minimum number of monochromatic copies of H over all k-colorings c : E(K n [k], De Silva et. al look at the maximum number of rainbow copies of H. One must consider rainbow copies of H since giving every edge of K n the same color clearly maximizes the number of monochromatic copies. Define rb k (H;n to be the maximum number of rainbow copies of H over all k-colorings c : E(K n [k]. This parameter is called the anti-ramsey multiplicity of H, and [1] investigates the behavior of this function for different graphs H. In this paper, we consider an arithmetic analog of anti-ramsey multiplicity thereby combining the problems raised in [1] with the arithmetic setting of [6]. We will focus entirely on the Sidon equation X + Y = Z + T. The Sidon equation measures the additive energy of a set. The additive energy of a set A Γ is the number of four tuples (a,b,c,d A 4 such that a+b = c+d. Typically it is written as E(A = {(a,b,c,d A 4 : a+b = c+d}. This fundamental parameter measures the additive structure of A, and for more on additive energy, see Tao and Vu [7]. Additive energy is perhaps one of the reasons why the Sidon equation is used as a first example in [6]. We would also like to remark that rainbow solutions to the Sidon equation were studied by Fox, Mahdian, and Radoičić []. They proved that in every 4-coloring of [n] where the smallest color classes has size at least n+1, there is at least one rainbow solution to the Sidon equation. This result 6 is also discussed in [] which surveys several problems on conditions ensuring a rainbow solution to an equation.

3 SinceweareinterestedinthemaximumnumberofrainbowsolutionstoX+Y = Z+T, we must take a moment to carefully describe how solutions are counted. First, since we are only counting rainbow solutions, we only care about solutions to X + Y = Z + T in which all of the terms are distinct. Additionally, we want to count solutions that can be obtained by interchanging values on the same side of the equation as being the same. With this in mind, we define a set of four distinct integers {x 1,x,x,x 4 } ( [n] 4 a Sidon 4-set if these integers form a solution to the Sidon equation X + Y = Z + T. Given a Sidon 4-set {x 1,x,x,x 4 }, we can determine exactly which pairs appear on each side of X+Y = Z+T. Without loss of generality, we may assume that x 1 is the largest among the x i s and x 4 is the smallest. It follows that x 1 +x 4 = x +x and again without loss of generality, we may assume x > x so x 1 > x > x > x 4. In short, given a Sidon 4-set {x 1,x,x,x 4 }, the two extreme values appear on one side of X +Y = Z +T, and the two middle values appear on the other side. Now we are ready to define the Ramsey function that is the focus of this work. Let n k 4 be integers. We define AR k X+Y=Z+T (n to be the maximum number of rainbow Sidon 4-sets over all colorings c : [n] [k]. It can be shown that the total number of Sidon 4-sets in [n] is exactly n 1 n 8 + 5n 1 θ where θ = 0 if n is even, and θ = 1 if n is odd. This immediately implies the upper 8 bound ARX+Y k n =Z+T (n 1 n 8 + 5n 1 for n k 4. A First Moment Method argument gives a lower bound of ( ( 1 k +O n O k k (n ARX+Y=Z+T k (n. Our first theorem improves both of these bounds. Theorem 1.1 For integers n k 4, ( k + θ k n O k (n AR k X+Y =Z+T (n ( k n +O k (n where θ = 1 if k is even, and θ = 1 if k is odd. 4 When k = 4, we can improve the upper bound using a different argument. Theorem 1. For n 4, n 96 O(n ARX+Y=Z+T 4 n (n 96 +O(n. The lower bound in Theorem 1. is a consequence of Theorem 1.1. Finding an asymptotic formula for ARX+Y=Z+T k (n is an open problem. The rest of this paper is organized as follows. In Sections and, we prove the upper bounds of Theorem 1.1 and Theorem 1., respectively. The lower bound is proved in Section 4. Some concluding remarks and further discussion is given in Section 5.

4 An Upper bound for k colors Key to our upper bound for k > 4 is the following lemma. It gives a lower bound for the number of Sidon 4-sets that contain a fixed pair. It will be applied to pairs that are monochromatic under a given coloring c. Lemma.1 Let n be a positive integer and let {b < a} ( [n]. Define fn ({b < a} to be the number of Sidon 4-sets {x 1,x,x,x 4 } ( [n] 4 with {a,b} {x1,x,x,x 4 }. Then f n satisfies f n ({b < a} n 4. Proof. We will consider two possibilities depending on the positioning of a and b within the equation x 1 +x 4 = x +x. Claim 1: If a+b = x i +x j, then the number of x i, x j [n] with x i < x j that satisfy this equation is at least { a+b 1 1 if a+b n+1, n a+b 1 1 if a+b > n+1. Proof of Claim 1: First suppose a+b n+1. Let x i = m and x j = a+b m where 1 m a+b 1. Since a + b n + 1, the integers x i and x j are in [n] for all m in the specified range. Since we must exclude x i = b, x j = a (a, b, x i, and x j must all be distinct to be a Sidon 4-set, we obtain that the total amount of possible values for x i,x j is at least a+b 1 1. Now suppose a + b > n + 1. Let x i = a + b n + m and x j = n m where 0 m n a+b 1. Since a+b > n+1, we have n m x i < x j n for all m in the specified range. As before, the solution x i = b and x j = a must be excluded. Here we obtain that the total amount of possible values for x i and x j is at least a+b 1 a+b 1 n n 1. Claim : If a+x i = b+x j, then the number of x i,x j [n] with x i < x j that satisfy this equation is at least n (a b. Proof of Claim : Note that a+x i = b+x j implies a b = x j x i. Since b < a, we have that a b > 0. Let x i = m and x j = a b+m. The range of m for which we have a valid solution is 1 m n (a b. However, we also require that {a,b} {x i,x j } = and so the solutions (x i,x j = (b a,b, (x i,x j = (b,a, and (x i,x j = (a,a b must all be excluded. Thus, we obtain the number x i, x j that satisfy the equation a+x i = b+x j and all other constraints is at least n (a b. These two possibilities (a + b = x i + x j and a + x i = b + x j are disjoint and cover all possible positions for a and b. A lower bound on the number of Sidon 4- sets {x 1,x,x,x 4 } ( [n] 4 with {a,b} {x1,x,x,x 4 } is obtained by combining these 4

5 two cases. So we have that if a+b n+1, then the amount of Sidon 4-sets that contain a and b is at least a+b 1 1+n (a b n a + b 11 n 4. If a+b > n+1, then the amount of Sidon 4-sets that contain a and b is at least a+b 1 n 1+n (a b n a + b 9 n 4. Theorem. For integers n k 4, ( 1 ARX+Y=Z+T k (n 1 1 n +O k (n. 4k Proof. Let c : [n] [k] be a k-coloring of [n]. Let X i be the integers assigned color i by c. Let M ( [n] be the set of all pairs {b < a} which are monochromatic under c, i.e., c(a = c(b. Then M = k ( Xi = 1 i=1 k X i 1 i=1 k X i 1 k n k = n k n. (1 Let f n ({b < a} be the number of Sidon 4-sets in [n] that contain {b < a}. By Lemma.1, f n ({b < a} n 4. ( The sum {b<a} M f n({b < a} counts the number of Sidon 4-sets that contain at least one monochromatic pair. A given Sidon 4-set is counted at most six times by this sum since there are ( 4 ways to choose a pair from a Sidon 4-set. In fact, the only Sidon 4-sets that will be counted six times in this sum are those which are monochromatic under c. All others will be counted at most three times. Regardless, we have that the number of Sidon 4-sets that are not rainbow under c is at least 1 f n ({b < a} k n {b<a} M {b<a} M where we have used both (1 and (. An Upper Bound for four colors i=1 For k = 4, the upper bound of Theorem. gives AR 4 X+Y=Z+T (n ( n +O(n. (n 4 = n 4k O k(n 5

6 In the special case that k = 4, we can obtain a better upper bound with a different argument based on additive energy. Let A 1,A,...,A t be finite sets of integers and define E t (A 1,A,...,A t = {(a 1,a,...,a t A 1 A A t : a 1 +a + +a t = 0}. For integers n m, write [n,m] for the interval For a finite set J Z with j elements, let {n,n+1,n+,...,m}. I(J = [ j/, j/ ]. Note that I(J depends only on the cardinality of J. A key ingredient in the proof of our upper bound is the following result of Lev [5]. Theorem.1 (Lev [5] Let t be an integer. For any finite sets A 1,A,...,A t Z, E t (A 1,A,...,A t E t (I(A 1,I(A,...,I(A t. The main idea is to apply Theorem.1 with t = 4, where A 1,A,A,A 4 are color classes of a coloring c : [n] [4]. Before using Theorem.1, we need a few lemmas. For finite sets A,B Z and an integer m, let r A+B (m = {(a,b A B : a+b = m}. Lemma. Let 1 α β be integers. If A = [ α,α] and B = [ β,β], then α+1 if m β α, r A+B (m = β +α+1 m if β α m α+β, 0 otherwise. In particular, r A+B (m β +α+1 m whenever m α+β. Proof. For any m with m β α, we can write m = j +(m j where j [ α,α]. The term m j is in B since if m β α and j α, then m j m + j β α+α = β. This shows that r A+B (m = α+1 whenever m β α. Now suppose β α m α+β, say m = β α+l for some l {0,1,...,α}. Then m = β α+l = ( α+l +t+(β t ( for t {0,1,...,α l}. We now check that for each such t, we have α + l + t A and β t B. Since α β and 0 l α, β β α β α+l = β (α l β t β 6

7 so β t β hence β t B. Similarly, α α+l+t α+l+α l = α so α+l+t A. Therefore, in (, the term α+l+t belongs to A and β t belongs to B. Furthermore, this is all of the ways to write m as a sum of an integer in A and an integer in B. We conclude that for β α m α+β, r A+B (m = β +α+1 m. The proof is completed by noting that if m > α + β, then r A+B (m = 0, and A and B are symmetric about 0 so that r A+B (m = r A+B ( m. As for the assertion that r A+B (m β+α+1 m for m α+β, it is enough to check that β +α+1 m α+1 for m β α. An easy computation shows that these two inequalities are equivalent. Lemma. If α is a positive integer and J = [ α,α], then { α+1 m if m α, r J+J (m = 0 otherwise. Proof. Apply Lemma. with α = β. Lemma.4 Let α 1,α,α,α 4 be positive integers such that α := α 1 + α + α + α 4 is divisible by 4. If A i = [ α i,α i ] and J = [ α/4,α/4], then for any integer m with m α/, r A1 +A (m+r A +A 4 (m r J+J (m. Proof. Let m be an integer with m α. By Lemma., r A1 +A (m+r A +A 4 (m α 1 +α +1 m +α +α 4 +1 m = (α/+1 m = r J+J (m. For the last equality, we have used Lemma. with J = [ α/4,α/4]. Lemma.5 Let α 1,α,α,α 4 be positive integers such that α := α 1 + α + α + α 4 is divisible by 4. If A i = [ α i,α i ] and J = [ α/4,α/4], then r A1 +A (mr A +A 4 (m m Z α m= α Proof. First we show that if m > α, then the product r A1 +A (mr A +A 4 (m r J+J (m. must be 0. If r A1 +A (m 0 and r A +A 4 (m 0, then by Lemma., m α 1 +α and m α +α 4. 7

8 Adding the two inequalities together gives m α 1+α +α +α 4 and so m α. Thus, r A1 +A (mr A +A 4 (m = m Z α m= α r A1 +A (mr A +A 4 (m. By Lemma.4, for any m with m α, we have r A 1 +A (m +r A +A 4 (m r J+J (m. Thus, the product r A1 +A (mr A +A 4 (m is at most r J+J (m. Since this holds for all m with m α, r A1 +A (mr A +A 4 (m = m Z α m= α which completes the proof of the lemma. r A1 +A (mr A +A 4 (m Lemma.6 If α is a positive integer and J = [ α,α], then E 4 (J,J,J,J = 16α +8α + 14α +1. α m= α r J+J (m Proof. We must count the number of 4-tuples (x 1,x,x,x 4 with α x i α and x 1 +x +x +x 4 = 0. For an integer m with 0 m α, we have r J+J (m = α +1 m by Lemma.. The number of 4-tuples (x 1,x,x,x 4 with α x i α and x 1 +x +x +x 4 = 0 is α m= α r J+J (mr J+J ( m = r J+J (0 + = (α+1 + = 16α α m=1 α m=1 r J+J (m +8α + 14α +1. (α+1 m Theorem.7 The function ARX+Y=Z+T 4 (n satisfies ARX+Y=Z+T 4 n (n 96 +O(n. Proof. First we assume that n is divisible by 8. An easy monotonicity argument will complete the proof for all n. Suppose c : [n] {1,,,4} is a 4-coloring of [n]. Let X i be the integers assigned color i by c and X i = c i n. The number of rainbow solutions to X + Y = Z + T is exactly N(c := E 4 (X 1,X, X, X 4 +E 4 (X 1,X, X, X 4 +E 4 (X 1,X 4, X, X. 8

9 By Theorem.1, N(c E 4 (I(X 1,I(X,I( X,I( X 4 +E 4 (I(X 1,I(X,I( X,I( X 4 + E 4 (I(X 1,I(X 4,I( X,I( X. We will show that each of the terms on the right hand side is at most n 96 +O(n. For 1 i 4, I(±X i = [ c i n/, c i n/ ]. Wealsohavethatc 1 +c +c +c 4 1. Assumethateach c in isaninteger. LetA 1 = I(X 1, A = I(X, A = I( X, A 4 = I( X 4, and J = [ n/8,n/8]. By Lemmas.5 and.6 E 4 (A 1,A,A,A 4 = m Z r A1 +A (mr A +A 4 ( m = m Zr A1 +A (mr A +A 4 (m n/4 m= n/4 r J+J (m = E 4 (J,J,J,J = n 96 +O(n. We apply this same estimate to E 4 (X 1,X, X, X 4 and E 4 (X 1,X 4, X, X to obtain N(c n 96 +O(n. If the c in are not integers, we can still apply the above argument but now J must be replaced with J = [ n/8 1,n/8 + 1]. Nevertheless, we still have E 4 (J,J,J,J n + 96 O(n as E 4 (J,J,J,J increases by O(n when the interval J is increases from [ n/8,n/8] to [ n/8 1,n/8+1]. If n is not divisible by 8, then let l be the smallest integer for which n+l is divisible by 8 (so 1 l 7. By monotonicity, AR 4 X+Y=Z+T(n AR 4 X+Y=Z+T(n+l (n+l 96 +O((n+l = n 96 +O(n. 4 A Lower Bound for k colors In this section we prove a lower bound on ARX+Y=S+T k (n for k 4. We will need two lemmas before proving the lower bound. In this section, we continue to write r A+B (m = {(a,b A B : a+b = m}. Lemma 4.1 Let 1 i < j k be integers and let n be a positive integer that is divisible by k. If X i = {m [n] : m i(mod k} and X j = {m [n] : m j(mod k}, then { t+1 if 0 t n r Xi +X j (i+j +tk 1, k n 1 t if n t n. k k k If l i+j(mod k, then r Xi +X j (l = 0. 9

10 Proof. First note that since k divides n, X i = {i,i+k,i+k,...,i+n k} and X j = {j,j +k,j +k,...,j +n k}. If l = a+b for some a X i and b X j, then l i+j(mod k. Thus, r Xi +X j (l = 0 whenever l i+j(mod k. This proves the last assertion of the lemma. Let t be an integer with 0 t n 1. We claim that for each α {0,1,...,t}, we k get i+j +tk = (i+αk+(j +(t αk where i+αk X i and j +(t αk X j. The inequality i i+αk i+tk i+ k 1 k = i+n k shows that i+αk X i for each α {0,1,...,t}. Similarly, j j +(t αk j +tk j + k 1 k = j +n k shows that j +αk X j for each α {0,1,...,t}. Consequently, r Xi +X j (i+j +tk t+1 whenever t {0,1,..., n 1}. k Now let t be an integer with n t n. Write t = n β where β n. For k k k k each α {1,,...,β 1}, we can write i+j +tk = i+j + k β k = ( i+ ( k α k + j + k (β α k. We claim that i+ α k X k i and j + (β α k X k j. Now i+k = i+n k 1 k i+n (β 1k i+ k α k i+n k where we have used the inequalities β n, α β 1, and α 1. We conclude that for k each α {1,,...,β 1}, the term i+ α k is in X k i. Similarly, j +k = j + k k 1 k j + j + k 1 k = j +n k k (β 1 k j + k (β α k shows that j + k (β α k is in X j for each α {1,,...,β 1}. Therefore, Since t = n k β, we have β 1 = n k r Xi +X j (i+j +tk β 1. t 1 and this completes the proof of the lemma. For the next lemma we will count Sidon 4-sets in Z k. A Sidon 4-set in Z k is a set of four distinct elements α, β, γ, δ Z k such that α+β γ +δ(mod k. We will denote such a 4-set by {α+β γ +δ}. The reason we cannot simply write {α,β,γ,δ} is that in Z k, four distinct residues may lead to more than one solution to the Sidon equation. For example, in Z 4, 10

11 1+ +4(mod 4 and 1+4 +(mod 4. ThisdoesnotoccurinZbecauseoftheorderingoftheintegers. LetS(kbethecollection of all Sidon 4-sets in Z k. Finishing off the example of k = 4, it is easily seen that S(4 = {{1+ +4},{1+4 +}} (4 and so S(4 =. We are now ready to state and prove the next lemma. Lemma 4. Let k 4 be an integer. If S(k is the family of all Sidon 4-sets in Z k, then S(k = k 8 k +θk where θ = 1 if k is even, and θ = if k is odd. 8 Proof. For this lemma, we will write a b for a b(mod k. Let us first assume that k is even. Where this will come into play is that when k is even, the congruence X b will have exactly two solutions when b is even, and no solutions when b is odd. First we choose a pair {x 1,x } ( Z k (. This can be done in k ways and this pair will be one side of the equation X +Y Z +T. Our counting from this point forward depends on if x 1 +x is even or odd when viewed as an integer. Case 1: x 1 +x is even If x 1 +x is even, then the congruence X x 1 +x has exactly two solutions, say y 1 and y. Note that no y i can be the same as an x i for if, say y 1 x 1, then from y 1 +y 1 x 1 +x we get x y 1 x 1 contradicting the way x 1 and x have been chosen. Therefore, in the case that x 1 +x is even, there are k 4 choices for x for which the unique x 4 satisfying x 1 +x x +x 4 will have the property that all of x 1, x, x, and x 4 are distinct. We conclude that {x 1 +x x +x 4 } is indeed a Sidon 4-set. This Sidon 4-set is counted exactly four times in this way: we could have chosen x or x 4 after having chosen the pair {x 1,x }, and we could have also started by choosing the pair {x,x 4 } instead. When k is even, the number of pairs {x 1,x } for which x 1 +x is even is exactly k 1 k t=1 t = k (this can be seen by looking 4 at the diagonals in a Cayley table for Z k. Altogether, we have a count of ( k k (k Sidon 4-sets {x 1 +x x +x 4 } where x 1 +x is even. Case : x 1 +x is odd If x 1 +x is odd, then X x 1 +x has no solution since gcd(,k does not divide x 1 +x. Nowtherewill bek choicesforx andtheuniquex 4 satisfying x 1 +x x +x 4 11

12 will have the property that {x 1,x,x,x 4 } is a 4-set. There are k t=1 (t 1 = k 4 pairs {x 1,x } for which x 1 +x is odd. This gives a count of ( k (k 4 4 Sidon 4-sets {x 1 +x x +x 4 } where x 1 +x is odd. Combining the two cases, there are exactly ( k k (k (k + (k 4 = k 4 8 k + k Sidon 4-sets in Z k when k is even. When k is odd, a similar counting argument can be done. The key difference is that for any pair {x 1,x }, the congruence X x 1 + x has exactly one solution since gcd(k, = 1 always divides x 1 +x. This unique solutionmust beavoided when choosing x and so there will be k choices for x. The rest of the counting is similar to as before and we obtain Sidon 4-sets in Z k when k is odd. ( k (k 4 = k 8 k + k 8 Theorem 4. Let n k 4 be integers and assume that n is divisible by k. If S(k is the family of all Sidon 4-sets in Z k, then ( n ARX+Y=Z+T k (n S(k k O k(n. Proof. Let n k 4 be integers where k divides n. Define the coloring c : [n] [k] by c(i = i(mod k where we use residues in the set {1,,...,k}. The number of rainbow Sidon 4-sets under c is n l=1 1 i<j<s<t k ( rxi +X j (lr Xs+X t (l+r Xi +X s (lr Xj +X t (l+r Xi +X t (lr Xj +X s (l (5 where X i = {m [n] : m i(mod k}. To see this, observe that if x 1 +x = x +x 4 is a Sidon 4-set that is rainbow, then there are distinct colors 1 i < j < s < t k with {c(x 1,c(x,c(x,c(x 4 } = {i,j,s,t}. ThisrainbowSidon4-setiscountedexactlyoncebythesum(5preciselywhenl = x 1 +x, and by only one of the terms in the sum r Xi +X j (lr Xs+X t (l+r Xi +X s (lr Xj +X t (l+r Xi +X t (lr Xj +X s (l. (6 The unique nonzero term depends on which two colors appear on the same side of the equation x 1 +x = x +x 4. For instance, if colors i and j appear on the same side, then the first term in (6 is the one that counts {x 1,x,x,x 4 }. 1

13 Fix an l [n] and four distinct colors i,j,s,t. By Lemma 4.1, the product r Xi +X j (lr Xs+X t (l is not zero if and only if l i + j(mod k and l s + t(mod k. This clearly implies i+j s+t(mod k and so {i+j s+t} is Sidon 4-set in Z k. For u {0,1,...,k 1}, let S(k, u be the Sidon 4-sets {α+β γ +δ} S(k for which α+β u(mod k. The collection {S(k,u : 0 u k 1} forms a partition of S(k. Since r Xi +X j (l 0 if only if l i+j(mod k, (5 can be rewritten as S := n k 1 l=0 k u=1 {α+β γ+δ} S(k,u r Xα+X β (u+klr Xγ+X δ (u+kl. In order to use Lemma 4.1, we split this sum into two sums S 1 and S where S S 1 +S. Define n k 1 k S 1 := r Xα+X β (u+klr Xγ+X δ (u+kl and By Lemma 4.1, S 1 n k 1 l=0 k S := l=0 n k l= n k u=1 {α+β γ+δ} S(k,u k u=1 {α+β γ+δ} S(k,u u=1 {α+β γ+δ} S(k,u n k 1 (l+1 l=0 r Xα+X β (u+klr Xγ+X δ (u+kl. ( n S(k (l+1 = S(k k O k(n. (7 A similar application of Lemma 4.1 gives ( n S S(k k O k(n. (8 Combining (7 and (8, we have ( n S S 1 +S S(k k O k(n which tells us that the number of rainbow Sidon 4-sets under the coloring c is at least the right hand side of (9. Corollary 4.4 For integers n k 4, the function ARX+Y=Z+T k (n satisfies ( 1 ARX+Y=Z+T k (n 1 1 k + θ n O k k (n (9 where θ = 1 if k is even, and θ = 1 4 if k is odd. 1

14 Proof. First assume that n is divisible by k. By Theorem 4. and Lemma 4., ( ( k ARX+Y=Z+T k (n 8 k n +γk k O k(n. where γ = 1 if k is even and γ = if k is odd. 8 If n is not divisible by k, then choose r [k 1] so that n r is divisible by k. We then have by monotonicity, ( ( k ARX+Y=Z+T k (n ARk X+Y=Z+T (n r 8 k (n r +γk O k k (n. The lower order term can be absorbed into the O k (n error term so we get ( ( k ARX+Y=Z+T(n k 8 k n +γk k O k(n ( 1 = 1 1 k + θ n O k k (n in either case. Here θ = 1 if k is even, and θ = 1 4 if k is odd. 5 Concluding Remarks In this paper we studied the anti-ramsey function ARX+Y=Z+T k (n which concerns colorings of [n]. One could also consider colorings of Z n. Write ARX+Y Z+T k (Z n for the maximum number of rainbow solutions to X +Y Z +T(mod n over all k-colorings c : Z n [k]. As in the case of [n], we count solutions that only differ by ordering as the same. This is discussed in detail prior to Lemma 4.. Now by Lemma 4., AR k X+Y Z+T (Z n n 8 n +θn where θ = 1 if n is even, and θ = if n is odd. When k = 4, it is easy to improve this 8 upper bound as follows. Let c : Z n [4] be a coloring of Z n and let X i be the elements of Z n assigned color i by c. The number of rainbow solutions to the Sidon equation X +Y Z +T(mod n where colors 1 and appear on the same side is at most min{ X 1 X X, X 1 X X 4, X 1 X X 4, X X X 4 }. (10 Indeed, once we have chosen three values for the four variables X, Y, Z, and T, the last variable is uniquely determined. Since X 1 + X + X + X 4 = n, (10 is at most n. 64 There aretwo other possible ways toobtainarainbow solutiontox+y Z+T(modn. One is where colors 1 and appear on the same side, and the other is where colors 1 and 4 appear on the same side. This gives the upper bound AR 4 X+Y Z+T(Z n n

15 As for a lower bound, a natural idea is to try the same coloring that is used to prove Theorem 4.. It turns out that this is not more difficult if we consider arbitrary k 4, nevertheless we restrict to k = 4 for simplicity. Define the coloring c : Z n [4] by c(i = i(mod 4 where we use residues in {1,,,4} for the colors. If n is not divisible by 4, then this coloring may not be well defined! A simple example is when n = 5 where c(5 = 1, andc(10 =, however, 5and10arethesame element ofz 5. Anobvious wayto fix this is to fix equivalence class representatives, say Z n = {1,,...,n}. Unfortunately this does not solve the problem as we still require the arithmetic in Z n when finding solutions to X +Y Z + T(mod n. To proceed further, let us now assume that n is divisible by 4 and so the coloring c will be well defined and will not depend on how we represent the elements of Z n. It is now straightforward to adapt Lemma 4.1 to the Z n case. For 1 i < j 4, we would have r Xi +X j (i+j +4t = n 4 for all t {0,1,..., n 4 1}, and r X i +X j (l = 0 if l i + j(mod 4. The proof of this follows along the same lines as the proof of Lemma 4.1, except now i+j +4t (i+4α+(j +4(t α(mod n for all α {1,..., n }. One then obtains the lower bound 4 AR 4 X+Y Z+T (Z n = = n 4 1 l=0 n u=1 {α+β γ+δ} S(4,u r Xα+X β (u+4lr Xγ+X δ (u+4l r X1 +X (+4lr X +X 4 (+4l+r X1 +X 4 (1+4lr X +X (1+4l l=0 n 4 1 ( (n l= = n again, assuming n is divisible by 4. When k = 4, determining an asymptotic formula for the number of rainbow solutions to the Sidon equation in [n] or Z n would certainly be interesting. Additionally, improving the upper bound ( 1 ARX+Y=Z+T k (n 1 1 n +O k (n 4k seems possible. Using the methods of this paper, one might be able to improve the 1 to 1, but we believe the lower bound is closer to the truth and so any significant 4k 1k improvement may require some new ideas. 6 Acknowledgment The authors would like to thank Michael Tait for suggesting the problem that is the focus of this paper and for some insightful comments. 15

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