1 Introduction and historical background
Let denote the set of positive integers. A triple is called a Schur triple if its entries satisfy the equation . The set of all positive integers up to will be denoted by . A coloring of is a map for some finite set of colors. For example, a map is a -coloring. We say that a Schur triple is monochromatic (with respect to a given coloring) if all of its entries have been assigned the same color; we will abbreviate “monochromatic Schur triple” by MST.
With these notations, one can ask questions like: given and a coloring of , how many MSTs are there in ? Let us denote this number as follows:
For our purposes, two Schur triples and are considered distinct if . We emphasize this convention since sometimes in the literature these two triples are counted only once, which is equivalent to imposing the extra condition . For example, there are four monochromatic Schur triples on when and are colored red and are colored blue, namely , , , and . We will use a short-hand notation for -colorings, namely as words on the alphabet : the -th letter is if the integer is colored red and if it is blue. So the above -coloring would be denoted by . We will also make use of the power notation for words, e.g., .
The namesake of the triples in this work refers to Issai Schur [Schur17], who in 1917 studied a modular version of Fermat’s last theorem (first formulated and proved by Leonard Dickson). In order to give a simpler proof of the theorem, Schur introduced a Hilfssatz confirming the existence of a least positive integer such that for any -coloring of an MST exists (this is nowadays known as Schur’s theorem). In 1927, Van der Waerden [Vanderwaarden27] generalized this result to monochromatic arithmetic progressions of any length . Then in 1928, Ramsey proved his eponymous theorem, showing the existence of a least positive integer such that every edge-coloring of a complete graph on vertices, with the colors red and blue, admits either a complete red subgraph or a complete blue subgraph. However, a real increase in the popularity of these kinds of Ramsey-theoretic problems came with the rediscovery of Ramsey’s theorem in a 1935 paper of Erdős and Szekeres [ErdosSzekeres35], which ultimately led to a simpler proof of Schur’s theorem, indicating their close connections. For the curious reader, this rich history is beautifully depicted in a book by Landman and Robertson [LandmanRobertson04].
We now arrive at a point of more than just questions of existence. In 1959, Alan Goodman [Goodman59] studied the minimum number of monochromatic triangles under a 2-edge coloring of a complete graph on vertices. Then in 1996, Graham, Rödl, and Ruciński [GrahamRodlRucinski96] found it natural to extend the problem of “determining the minimum number under any 2-coloring” to Schur triples. In fact, Graham offered a prize of 100 USD for an answer to such a question; it has subsequently been successfully answered many times over, in an asymptotic sense. In order to give some more context to this problem, we first introduce some additional notation.
We start by wondering about what we can say about the number of MSTs on if we do not prescribe a particular coloring. It is not difficult to calculate that there are exactly Schur triples on . Trivially, this yields an upper bound for the number of MSTs, which can be achieved by coloring all numbers with the same color. This is the reason why it is more natural (and more interesting!) to ask for a lower bound for , that is: for given , what is the “best” lower bound for the number of MSTs regardless of the choice of coloring? Of course, is a trivial such lower bound, but we are aiming for something sharp, in the sense that for each there exists a coloring for which this bound is actually attained.
Differently stated, we are looking for the minimal number of monochromatic Schur triples among all possible colorings of :
For example, for , one cannot avoid the occurrence of monochromatic Schur triples, but there exists a -coloring for which only a single such triple occurs, namely the triple for the coloring . Hence, .
As mentioned before, this problem was only studied from an asymptotic point of view: Robertson and Zeilberger [RobertsonZeilberger98] was first to give the lower bound as (and consequently won Graham’s cash prize), where it has to be noted that they count only Schur triples with the condition imposed. This lower bound was independently confirmed by Datskovsky [Datskovsky03], Schoen [Schoen99], and Thanatipanonda [Thanatipanonda09]. Schoen also provided a proof of an “optimal” coloring of that would give such a minimum number, and such a coloring is what we assume later in this paper. The asymptotic lower bounds for the generalized Schur triples case for is as , without the requirement of . This was conjectured by Thanatipanonda [Thanatipanonda09] and Butler, Costello, and Graham [ButlerCostelloGraham10], and subsequently proven in 2017 by Thanatipanonda and Wong [ThanatipanondaWong17].
In this paper, we take a slightly different approach by using known computer algebra techniques and creative simplifications to develop exact formulas for the minimum number of such triples (in both the Schur triples case and the generalized Schur triples case) and give an analysis of the transitional behavior between the cases. Thus, in order to keep some consistency for comparison, we will remove the assumption of when counting MSTs. In this way, we can explain why the behavior of the minimum number of triples jumps when moving from the case to the case (note that the above asymptotic formula does not specialize to the expected prefactor when is substituted).
The overall plan is to systematically exploit the full force of symbolic computation and perform a complete analysis of determining the minimum number of monochromatic triples in both the discrete context and the continuous context . This requires three courses of a mathematical meal. We serve an appetizer in Section 2, showing how to derive an exact formula for the minimum in the classic Schur triple case (corresponding to in the general equation). This sets us up for the main course in Section 3, where we perform a full analysis for , illustrating that a global minimum can always be found. Interesting transitional behaviors occur at many locations for and one key transition occurs at . Admittedly, this course may be a bit difficult to swallow, and we hope that the reader will not suffer from indigestion. For dessert, we follow the procedure described in Section 2, and illustrate how it can systematically produce (ostensibly, an infinite number) of exact formulas for the minimum number of generalized Schur triples. Accordingly, in Section LABEL:sec:discrete, we leave the reader with exact formulas for the minimum number of generalized Schur triples for , and , with the hope that s/he will leave satisfied.
For the reader’s convenience, all computations and diagrams are in the Mathematica notebook [KoutschanWong19] that accompanies this paper, freely available at the first author’s website.
2 Exact lower bound for monochromatic Schur triples
It has been shown previously [RobertsonZeilberger98, Schoen99] that for fixed the number is minimized when consists of three blocks of numbers with the same color (“runs”), i.e., when is of the form , where and are approximately and , respectively. In this section, we derive exact expressions for the optimal choice of and , as well as for the corresponding minimum .
Let be such that . Moreover, assume that the inequalities and hold. Then the number of monochromatic Schur triples on under the coloring , denoted by , is exactly
In Figure 1 the situation is depicted for , , and . One sees that the dots representing the MSTs are arranged in four regions of right triangular shape. The triangles arise as follows:
The dots in the lower left corner correspond to red MSTs all of whose components are taken from the first block of red numbers; hence there are dots in the first row of this triangle.
The triangle in the center contains all blue MSTs, whose first two components satisfy the inequalities , , and . Note that such MSTs only exist if (for and the second term in (3) vanishes and the formula is still correct). The number of dots on each side is therefore .
The two triangles in the upper left and lower right corners correspond to red MSTs, whose first two entries belong to different blocks of red numbers. By symmetry they have the same shape and they have dots on their sides. Here we use the condition , because otherwise these two regions would no longer be triangles and we would be counting different things beyond the scope of our assumptions.
Adding up the contributions from these three cases, one obtains the claimed formula. ∎
The optimal values for and are easily derived using the techniques of multivariable calculus, once the form is assumed: by letting go to infinity and by scaling the square to the unit square , we see that the portion of pairs for which is an MST among all pairs in equals the area of a certain region in the unit square; for example, see the shaded regions in Figure 1. In this limit process, the integers and turn into real numbers satisfying . According to (3) the area of the shaded region in Figure 1 is given by the formula
Equating the gradient
to zero, one immediately gets the location of the minimum .
For fixed , the integers and that minimize the function are given by
Strictly speaking, we prove the minimality of the function under the additional assumption from Lemma 1. The fact that this is also the global minimum for all follows as a special case from the more general discussion as described in the proof of Lemma LABEL:lem:min.
The statement is proven by case distinction into cases, according to the remainder modulo . Here we show details for the case , and the remaining cases can be similarly verified with a computer; for these cases we refer the reader to the accompanying electronic material [KoutschanWong19].
By setting we can eliminate the floors from the definitions of and ; we obtain and . Our goal is to show that among all integers the expression is minimal for . Using (3) one gets
The stated goal is equivalent to showing that the polynomial
is nonnegative for all . Such a task can, in principle, be routinely executed by cylindrical algebraic decomposition (CAD) [Collins75]. In this method, the variables and are treated as real variables, which causes some problems in the present application. The reason is that does not hold for all . The situation is depicted in Figure 2, where the ellipse represents the zero set of and its inside those values for which the polynomial is negative. To our relief, we see that no integer lattice points lie inside the ellipse, since such points would be counterexamples to our claim.
Our strategy now is the following: we prove that for all integer points that are close to , e.g., for all with and . These points are shown in Figure 2, with the respective value of attached to them. In particular, we see that the minimum is attained several times, namely on the three points that lie exactly on the boundary of the ellipse.
Then we invoke cylindrical algebraic decomposition on the formula
which states that if the point lies outside the square that we have already considered, then holds. Calling the Mathematica command CylindricalDecomposition with input (4), we immediately get True. ∎
We are now ready to state the main theorem of this section, which is an exact formula for the minimal number of MSTs for any -coloring of . Apart from the asymptotic results mentioned in Section 1, there is only one paper [Schoen99] where a similar result is stated, but only for the case and for Schur triples with . In contrast, we consider all and our formula holds for all .
The minimal number of monochromatic Schur triples that can be attained under any -coloring of is
As in Lemma 2, we argue by case distinction , . Using and from the lemma, we obtain the following values for :
One easily observes that in each case, the result is of the form , where holds for all . Hence the claimed formula follows. ∎
The first terms of the sequence are
We have added this sequence to the Online Encyclopedia of Integer Sequences [Sloane] under the number A321195.
3 Asymptotic lower bound for generalized Schur triples
We now turn to generalized Schur triples, i.e., triples subject to for some parameter , as studied by Thanatipanonda and Wong [ThanatipanondaWong17]. Here, we allow to be even more general, i.e., . Consequently, we have to adapt the definition of generalized Schur triples: we use the condition . The case does not add new aspects to the analysis, as it can be transformed to the case by exchanging the roles of and and by changing the floor function to a ceiling.
Again, we choose to use the assumption that the minimal number of monochromatic generalized Schur triples (MGSTs) occurs at a coloring in the form of three blocks . We justify using this assumption with the experimental evidence of Butler, Costello, and Graham [ButlerCostelloGraham10] (who argued for the generalized Schur triple case ) and adapting the intuition in the argument of Schoen [Schoen99] (who only argued for the Schur triple case ).
We would like to know for which choice of and (depending on and ) the minimum occurs. Similar to the previous section, we let go to infinity and correlate the number of MGSTs with the area of polygonal regions in the unit square. We then define a function that determines this area, and minimize it. Hence, throughout this section, and are real numbers with .
Figure 3 shows two situations for different choices of . In contrast to the previous section, we do a very careful case analysis and do not impose extra conditions on and as in Lemma 1, at the cost of introducing a “few” more case distinctions. The full case analysis for normal Schur triples then follows by specializing to in the resulting formulas.
In the process of analyzing the different cases, we encounter several conditions on . For our referencing convenience, we distinguish these conditions here using the following abbreviations:
In Figures LABEL:fig:pwregions1 and LABEL:fig:pwregions2, the lines that represent some of these conditions are depicted. They split the triangle into several regions, depending on the value of .
Let with and . Then the area of the region
is given by a piecewise defined function, where case distinctions have to be made. For the sake of brevity, only the first cases are listed below, since they will be the most important ones in the subsequent analysis; in fact they are sufficient to describe for . We label the region corresponding to the -th case as . They are expressed in terms of the conditions (5) (where overlines denote negations):
As can be seen in Figure 3, the region whose area we would like to determine is the union of several polygons. Let , , and denote the intervals that correspond to the different blocks of the coloring ( and being red and being blue). Then is allowed while is not. From this point on, we will refer to the case by . It is easy to see that we have to consider only seven cases: , , , , , , . The cases and are clearly impossible since contradicts . All other combinations of violate the monochromatic coloring condition.
In both parts of Figure 3, case corresponds to the triangle that touches the origin. The coordinates of its other two vertices are and , hence its area is . However, this is valid only for . If , then the point is above the line and so the top of the triangle is cut off. As a result, one obtains a quadrilateral with vertices , , , , whose area is given by .
The case is similar, with the difference being that the corresponding polygon disappears if ; in the right part of Figure 3 the polygon is present while in the left part it is not. The polygons , , and are characterized by comparably simple case distinctions, while and require a much more involved analysis. In Figure LABEL:fig:cases133, we present such an analysis for , and refer to the accompanying electronic material [KoutschanWong19] for .
What we have achieved so far is a representation of as a sum of seven piecewise functions. However, what is required is a representation of as a single piecewise function, since that will be needed for determining the location of the minimum.
The conditions that are used to characterize the different pieces in Figure LABEL:fig:cases133 (and in the remaining cases that have not been discussed explicitly), are listed in (5). In order to combine the seven piecewise functions, we need a common refinement of the regions on which they are defined. We start with the finest possible refinement, which is obtained by considering all logical combinations of and for . Using Mathematica’s simplification procedures, we remove those cases that contain contradictory combinations of conditions, such as for example. After this purging, we are left with a subdivision of the set
which is an infinite triangular prism, into polyhedral regions. Finally, we merge regions on which is defined by the same expression into a single region, yielding a representation of as a piecewise function defined by different expressions. Each of them is of the form where is a polynomial in of degree at most in each of the variables. For more details, and to see the definition of in its full glory, see the accompanying electronic material [KoutschanWong19]. ∎