On Monochromatic Pairs with Nondecreasing Diameters

Let n, m, r, t be positive integers and ∆ : [n] → [r]. We say ∆ is (m, r, t)permissible if there exist t disjoint m-sets B1, . . . , Bt contained in [n] for which (a) |∆(Bi)| = 1 for each i = 1, 2, . . . , t, (b) max(Bi) < min(Bi+1) for each i = 1, 2, . . . , t− 1, and (c) max(Bi)−min(Bi) 6 max(Bi+1)−max(Bi+1) for each i = 1, 2, . . . , t− 1. Let f(m, r, t) be the smallest such n so that all colorings ∆ are (m, r, t)-permissible. In this paper, we show that f(2, 2, t) = 5t− 4. Mathematics Subject Classifications: 05D10, 11B75, 11B50

We say the collection {B 1 , B 2 , . . ., B t } is permissible in ∆ if they satisfy the above conditions, thereby realizing ∆ as (m, r, t)-permissible.
The first condition requires the m-sets be monochromatic; note two m-sets can be associated with different colors.The second condition is a non-overlapping property establishing precedence between the m-sets, and the third condition requires the ranges of each m-set, called their diameters, form a nondecreasing sequence.
In this language the question posed by Bialostocki, Erdős, and Lefmann [2] is as follows: Question 2. Given positive integers m, r, and t, does there exist an integer n such that for all ∆ : [n] → [r], ∆ is (m, r, t)-permissible?If so, what is the minimum possible value for n?
The answer to the first question is yes; it follows from a result of van der Waerden involving arithmetic progressions [11].Bialostocki et al. define f (m, r, t) as the smallest integer n for which all r-colorings of [n] are (m, r, t)-permissible.Several infinite families of parameters have f (m, r, t) determined or bounded.Observation 3. Note that for all positive integers m, r, and t, f (m, r, t) mt since the union of t disjoint m-sets realizing a coloring as (m, r, t)-permissible must have cardinality no less than f (m, r, t).This inequality is an equality if and only if m = 1 or r = 1.Additionally f (m, r, 1) = (m − 1)r + 1 by the pigeonhole principle.
Based on the above observation, interest in this question focuses on parameter families in which m, r, and t are at least 2.
(b) For a fixed t, f (2, r, t) is linear in r.
In 2005 Grynkiewicz showed the lower bound of the inequality in Theorem 4(c) is an equality when r = 4.
Recent work has been done on generalizations of Question 2, as well as investigations into its relationship to a theorem by Erdős, Ginzburg, and Ziv (see [3,4,6,7,8,10] for examples).With possibly the only exceptions being the bounds given in Theorem 4(d) and (f), as well as Theorem 5(b), all recent work in evaluating f (m, r, t) focuses on parameter families with t = 2 and more recently t = 3 [1].In our work we compute values for f (2, 2, t) for arbitrarily large t which, as far as we can tell, is the first known set of actual values of f (m, r, t) with m, r 2 and arbitrarily large t, rather than asymptotic results.
Note that f (2, 2, t) has been computed when t 3; f (2, 2, 1) = 3 by Observation 3, f (2, 2, 2) = 7 by Theorem 4(a), and f (2, 2, 3) = 12 by Theorem 7. Additionally, f (2, 2, t) 6t − 3 by Theorem 4(f).Note that the upper bound given in Theorem 4(f) is not an attempt at a sharp bound.The inequality is found by using the pigeonhole principle and, in fact, guarantees the existence of a permissible collection of t 2-sets which all have the same diameter.
In this paper we prove the following: In Section 2 we introduce notation for the problem and present some constructions which are useful in later proofs for finding collections of permissible sets.In Section 3 we first prove that f (2, 2, t) 5t − 4 by presenting a coloring of [5t − 5] which is not (2, 2, t)permissible, then prove a weak upper bound on f (2, 2, t) which gives the existence of a permissible collection of sets with additional structure.We finish the section with a proof that all colorings of [5t − 4] with t 4 are (2, 2, t)-permissible by showing there does not exist a coloring of [5t − 4] which is not (2, 2, t)-permissible.

Definitions and Constructions
We begin this section with a series of definitions, terminology, and notation related to colorings with 2 colors (using the color set {a, b} rather than [2]) and the identification of collections of 2-sets (abbreviated pairs) which are permissible.
Let n be a positive integer and ∆ : [n] → {a, b}.At times we present ∆ as the string , where x j = ∆(j) for each j = i, i + 1, . . ., i + k − 1, is a string of length k in ∆.For x ∈ {a, b}, we use the abbreviation x n to denote the string xx • • • x (n times) and similarly define w n = ww • • • w (n times), where w is any word from the alphabet {a, b}.We say {i, i + 1, . . ., i we say the k-tuple is isolated.Specifically, a double is a 2-tuple and a triple is a 3-tuple.A string Note that a D 1 is a double and a triple contains two D 1 s and a D 2 .
We now define the alternating substring/triples partition of [n] with respect to ∆ (abbreviated the AST partition of ∆) and give the process for constructing it.As we see later, this partition yields a permissible collection of pairs which have diameters 1 or 2, which is instrumental going forward. 1. Find the maximum number of pairwise disjoint triples in ∆.Suppose that there are w of them; call them T 1 , T 2 , . . ., T w , and suppose the minimal element in T i is τ i for each i ∈ [w].We additionally require that for all i ∈ [w], if This ensures that we "frontload" the triples.In other words, we read through the string from left to right and define our triples in a greedy method.We define the collection of triples as T := {T 1 , T 2 , . . ., T w }.
Note that some of these sets may be empty if ∆ contains consecutive triples, or begins or ends with a triple.It follows that U ∪ T is a partition of [n].

3.
Partition each U i into consecutive alternating substrings of maximal length.Note that if there are s such maximal substrings in U i , then there will be s − 1 doubles contained in U i .Thus in total we have v + w + 1 such maximal alternating substrings , where we use the convention that an empty set U i corresponds to an empty substring.Thus the substrings S i partition the terms of w i=0 U i as v+w+1 i=1 S i .Let S = {S 1 , S 2 , . . ., S v+w+1 } be the collection of these alternating substrings, and for each Observe that S ∪ T is a partition of [n], and so we define (S, T ; v, w) as the AST partition of ∆.Since S ∪ T is a partition of the electronic journal of combinatorics 26(2) (2019), #P2.15 Example 10.Let ∆ 1 : [16] → {a, b} be given by abaaababaabaaaba.Then the AST partition of ∆ 1 has v = 1 and w = 2 with T = {{3, 4, 5}, {12, 13, 14}} and S = {{1, 2}, {6, 7, 8, 9}, {10, 11}, {15, 16}}.One can see this visually as Let ∆ 2 : [36] → {a, b} be given as abababababaababaaabaabbababbbaaaaab.Then the AST partition of ∆ 2 has v = 4 and w = 3 with the parts indicated below. ∆ In this manner, all alternating strings of length k contain a permissible collection of k/3 D 2 s.Definition 15.Let ∆ : [n] → {a, b} with AST partition (S, T ; v, w).Each of the w triples in T contain a D 2 , while each alternating substring in S contains a permissible collection of k/3 D 2 s.Hence ∆ has a permissible collection of w As previously indicated in Observation 14, we "frontload" when selecting D 2 s from an alternating substring by reading from left to right.We call this collection of D 2 s of ∆ the canonical D 2 s of ∆.
the electronic journal of combinatorics 26(2) (2019), #P2.15 Proof.Let (S, T ; v, w) be the AST partition of ∆.By Observation 13 if v + w t, then ∆ is t-permissible as it contains at least t disjoint D 1 s.Now suppose v + w t − 1. Recall that for each i ∈ [v +w +1], k i = |S i |, and define k i ∈ {0, 1, 2} so that So σ t − 2 3 , and since σ and t are integers, σ t.By choosing any t of the σ canonical D 2 s of ∆, we have a permissible collection of t pairs in ∆.So ∆ is (2, 2, t)-permissible and is realized by a permissible collection of pairs with diameter at most 2.

Colorings of [5t − 4] which are not (2, 2, t)-permissible
For the remainder of this section, let ∆ be a coloring of [5t − 4], (S, T ; v, w) be the AST partition of ∆, D be the set of canonical D 2 s of ∆, and σ = |D|.We now outline a series of properties that ∆ must satisfy to not be (2, 2, t)-permissible.We conclude this section by showing no coloring can satisfy all such conditions, which establishes f (2, 2, t) 5t − 4, and thus proves Theorem 8.
As it will be relevant in the proof of Theorem 8, take note that all of the following lemmas and observations are valid for t 2, and we indicate where appropriate what the largest diameter for a realization of permissibility is for a coloring.This will be leveraged in the final proof.We begin an observation about the number of canonical D 2 s in ∆ provided it is not (2, 2, t)-permissible.
So σ t and therefore ∆ contains at least t canonical D 2 s.Hence ∆ is (2, 2, t)-permissible with a realization whose largest diameter is 2.
Adapting the proof for Theorem 18 allows us to determine the congruence for |S i | for each i = 1, 2, . . ., v + w + 1.
Proof.By Lemma 20, we have t = v + w + 1 and therefore ∆ has t alternating substrings in its AST partition.Again for each i ∈ [t], define k i ∈ {0, 1, 2} so that k i ≡ k i mod 3.By Observation 19 we have σ = t − 1. Combining this with a similar computation to (1) we get Therefore exactly one term in this sum is 1 while the remaining terms are each 2. Hence ∆ is (2, 2, t)-permissible with a realization whose largest diameter is 2.
We now classify the k-tuples which may exist in ∆.
Proof.First, observe that if ∆ contains a k-tuple for some k 6, then ∆ has at least two consecutive triples in its AST partition.Hence there is an empty substring S i ∈ S for some i ∈ [v + w + 1], giving that k i ≡ 0 mod 3.So ∆ is (2, 2, t)-permissible by Lemma 21 with a realization whose largest diameter is 2 -a contradiction.Now suppose ∆ contains the isolated 4-tuple { , + 1, + 2, + 3} for some ∈ [n − 3].For each i ∈ [w], let T i be the set containing the smallest two elements in T i .It follows from Construction 9 that T j = { , + 1, + 2} for some j ∈ {1, 2, . . ., w} and + 3 does not belong to any double or triple identified in the AST partition.Then {P 1 , P 2 , . . ., P v , T 1 , T 2 , . . ., T w , { +2, +3}} is a permissible collection of v +w +1 doubles, and since v + w + 1 = t by Lemma 20, ∆ is (2, 2, t)-permissible with a realization whose largest diameter is 1 -a contradiction.
With the previous two lemmas, we now establish that ∆ can have at most one triple and, if ∆ contains a triple, give conditions on the lengths of the alternating substrings surrounding the triple.
Proof.Observe that ∆ cannot begin with a triple, end with a triple, or have two consecutive triples in its AST partition; otherwise one of its alternating substrings has length 0, which contradicts Lemma 21.Therefore every triple in the AST partition of ∆ is preceded and followed by a nonempty alternating substring.Furthermore by Lemma 21, it follows that if ∆ contains a triple in its AST partition, then ∆ must contain (1, τ, 2), (2, τ, 1), or (2, τ, 2).
Note that in each of the above alterations, each coloring was found to be (2, 2, t)permissible with a realization whose largest diameter is 2.
We now give an observation which classifies the possible final substring in ∆ and is useful in the arguments which follow.
With this in mind, since ∆ cannot end with a triple, the alternating substring which ends ∆ cannot have length exceeding 2. So k v+w+1 = 1 or k v+w+1 = 2 and thus ∆ must end with type (1) or (2). ) and ending with any of these three types, then ∆ is also (2, 2, t)-permissible.This will be relevant in the last subcase of this proof.
• (2, 1, 2).In this case ∆ ends with baaab, meaning the triples were not correctly identified in its AST partition.So ∆ cannot end with this type.
We conclude with the observation that, in this work, we heavily use the binary nature of 2-colorings for constructing the AST partition used in our arguments.Hence we do not believe this method will translate nicely to [r]-colorings with r 3 or the discovery of permissible m-sets with m 3.
Definition 11.Let (S, T ; v, w) be the AST partition of ∆ : [n] → {a, b} for some positive integer n.Observe that S ∪ T is a partition of [n] into sets of consecutive integers, and these v+2w+1 parts have an implied order.With this in mind, we say ∆ is of type γ, where γ = (γ 1 , γ 2 , . . ., γ v+2w+1 ) is a (v + 2w + 1)-tuple, which is an ordering of k 1 , k 2 , . . ., k v+w+1 and w copies of the symbol τ where the order of the ordinates reflects the relative order of the alternating substrings and triples in the AST partition of ∆.We say ∆ contains y = (y 1 , y 2 , . .., y ) for some and symbols y i , i ∈ [ ] if, for some j, y i = γ i+j for each i ∈ [ ]. Specifically, we say ∆ ends with y if ∆ contains y and j = v + 2w + 1 − .For convenience, we use a to mean "is congruent to a mod 3 and at least a".Example 12. Let ∆ 1 and ∆ 2 be the colorings from Example 10.Then ∆ 1 is type (2, τ, 4, 2, τ, 2) and ∆ 2 is type (11, 4, τ, 2, 2, 4, τ, 0, τ, 1, 2).We may also say ∆ 1 contains (τ, 1, 2, τ ) and ends with (τ, 2), while ∆ 2 contains (1, τ, 2) and ends with (τ, 1, 2).Observation 13.Let (S, T ; v, w) be the AST partition of ∆ : [n] → {a, b}.If v + w t, then ∆ contains w disjoint triples -each of which contain a double -and v disjoint doubles which are disjoint from the triples.Therefore ∆ contains v + w t disjoint D 1 s, which implies ∆ is (2, 2, t)-permissible.Observation 14.Let k > 0 and w : [k] → {a, b} be a coloring represented by an alternating string, i.e. w = abab . . .a if k is odd and w In a similar manner there are 8 canonical D 2 s in ∆ 2 , which are illustrated below.We prove Theorem 8 in two stages.First, we give what will be the sharp lower bound on f (2, 2, t), then produce a weaker upper bound by showing when n is large enough, there is a permissible collection of pairs with bounded diameters contained in any 2-coloring of[n].We conclude the section by establishing properties that a non-permissible coloring of [5t − 4] must have, then showing that no coloring satisfies all of the properties.We first show that f (2, 2, t) 5t − 4 by demonstrating the existence of a coloring of [5t − 5] which is not (2, 2, t)-permissible.This is derived by setting m = r = 2 in the following theorem.
Theorem 17.Let m, r, t 2 and ∆ be the r-coloring of [(mr + 1)(t − 1)] given by The previous observation allows us to classify the value of v + w in the AST partition of ∆ given it is not (2, 2, t)-permissible.Lemma 20.Let t 2. If ∆ is not (2, 2, t)-permissible, then v + w = t − 1. Therefore ∆ has exactly t alternating substrings in its AST partition.
(2)of of Theorem 8. Suppose ∆ is not (2, 2, t)-permissible.By Observation 24(a), ∆ must end with type (1) or(2).By Lemma 23, ∆ may contain at most 1 triple and if ∆ contains a triple, then (2, τ, 1) or (1, τ, 2) is contained in ∆.That said there are 12 types with which ∆ may end; see Figure1.We now show in each case above that ∆ is, in fact, (2, 2, t)-permissible and therefore prove Theorem 8. Without loss of generality, we assume ∆(5t − 4) = b.Suppose ∆ ends with type:• (τ, 2).Since ∆ cannot end with bbbab by Observation 24(a), we have ∆ ends with aaaab.Since t 2, there is a nonempty alternating substring S t−1 which precedes the triple.If the last letter in S t−1 is an a, then by the AST construction ∆ has two consecutive triples, which violates Lemma 23.So ∆ ends with baaaab, in which case we have an isolated quadruple in ∆, which contradicts Lemma 22.