Abstract
Let be positive integers. By definition, is the least positive integer such that, for any -coloring of the interval , there exists a monochromatic solution to . For , the numbers are classical Schur numbers. In this paper, we study the numbers for .
1. Introduction
Ramsey theory, an area that has seen a remarkable burst of research activity during the past twenty years, is the study of the preservation of properties under set partitions. In other words, given a particular set that has a property , is it true that whenever is partitioned into finitely many subsets, one of the subsets must also have property ? Ramsey theory is named after Frank Plumpton Ramsey and his theorem, which he proved in 1928. Ramsey stated his fundamental theorem in a general setting and applied it to formal logic. Schur [1] and van der Waerden [2] obtained similar results in number theory. Dilworth’s theorem [3] for partially ordered sets is another typical example. Ramsey’s theorem was applied to geometry by Erdős and Szekeres [4]. They also defined the Ramsey numbers and gave some upper and lower bounds for them. Ramsey-type theorems have applications in different branches of mathematics such as number theory, set theory, geometry, ergodic theory, and theoretical computer science. In [5], connections between the computation of van der Waerden numbers and propositional theories have been shown. Using these connections, they serve as benchmarks for solvers of satisfiability problems. For more on Ramsey theory and its applications, we refer to the book of Graham et al., [6], and the surveys of Nešetřil [7] and Rosta [8]. Somewhat surprisingly, Ramsey’s theorem was not the first theorem in the area now known as Ramsey theory. The result that is generally accepted to be the first Ramsey-type theorem is due to Schur [1] and it deals with colorings of the integers: if is partitioned into a finite number of classes, at least one partition class contains a solution to the equation .
Let us now go over the details of some definitions and notations from [9]. An interval is a set of the form , where are integers.
Definition 1. An -coloring of a set is a function , where .
In fact, an r-coloring of a set is a partition of into subsets , by associating the subset with the set .
Definition 2. A coloring is monochromatic on a set if is constant on .
It is often convenient to represent a particular 2-coloring of an interval as a string of 0s and 1s. For example, the coloring with and could be represented by the string 11100 or . As we mentioned before, one of the earliest results in Ramsey theory was proved by Issai Schur in 1916.
Theorem 1. (Schur’s Theorem). For any , there exists a least positive integer such that, for any -coloring of , there exists a monochromatic solution to .
The numbers are called the Schur numbers. The known Schur numbers are , , , and . Note that and in Schur’s theorem need not be distinct. If we require that and be distinct, the resulting Ramsey-type statement is also true.
Theorem 2. For , there exists a minimal integer such that every -coloring of admits a monochromatic solution to with and being distinct.
It is known that Schur’s theorem is a consequence of Ramsey’s theorem. There are a number of interesting results proved during the last years concerning Schur’s theorem and generalizations. A triple of natural numbers is called a Schur triple if and . Let be the minimum number of monochromatic Schur triples in any 2-coloring of . Graham et al. [10] found the lower bound . They used the Ramsey multiplicity result [11, 12], which says that in every 2-coloring of the edges of a complete graph on vertices, there are at least monochromatic triangles. Answering a question raised in [10], Robertson and Zeilberger [13] and, independently, Schoen [14] showed that . Robertson and Zeilberger found a 2-coloring with monochromatic Schur triples and formulated a conjecture on the minimum number of triples. Schoen showed that every extremal coloring looks like the Robertson–Zeilberger construction, and he used this result to find the exact number , for .
One can look at Schur’s theorem in terms of sum-free sets. A set is called sum-free if implies . The Schur function is defined as the maximum such that can be partitioned into sum-free sets. We mention here a generalization of Schur’s theorem for sum-free sets: if is finitely colored, there exists arbitrarily large finite set such that the sum-free set of A, is monochromatic. Note that Hindman’s theorem [15] gives the same result when is an infinite set. On the other hand, Alekseev and Savchev [16] considered a similar problem and proved that for every equinumerous 3-coloring of (i.e., a coloring in which different color classes have the same cardinality), the equation has a solution, with , and belonging to different color classes. Such solutions will be called rainbow solutions. Moreover, Schönhei̇m [17] proved that for every 3-coloring of , every color class has cardinality greater than and the equation has rainbow solutions. Moreover, he showed that is optimal.
We now recall another generalization of Schur’s theorem. Let represent the equation , where are variables.
Theorem 3. Let and, for , assume that . Then, there exists a least positive integer such that for every -coloring of , there is a solution to of color for some .
Just as Schur’s theorem, Theorem 3 follows from Ramsey’s theorem. The numbers are called the generalized Schur numbers. In [18], the authors determine 26 previously unknown values of and conjecture that for , .
Following a problem proposed in [9], we consider the monochromatic solutions to . For abbreviation, we write instead of .
Definition 3. For integers , let be the least positive integer such that, for any -coloring of , there exists a monochromatic solution to .
The existence of such monochromatic solutions is implied by Rado’s theorem.
Theorem 4. (Rado’s Theorem). For any , there exists such that for every -coloring of , there is a monochromatic solution to the linear equation , where for , if and only if some nonempty subset of the ’s sums to 0.
2. Exact Value of
In this section, we find the exact value of . The standard methodology for finding the exact value of any particular Ramsey-type number is to show that some number serves both as a lower bound and an upper bound. We may illustrate this phenomenon by a simple example. We will establish that by using the above method to prove that and .
To show that , it suffices to exhibit a 2-coloring of with no monochromatic solution to . One such coloring is the following: color the intervals and red and color the interval blue. It is easy to see that this coloring avoids any monochromatic solution to . For all possible solutions, see Table 1.
To show that , we must show that every 2-coloring of admits a monochromatic solution to . Using red and blue as the colors, assume, for a contradiction, that there exists a 2-coloring of with no monochromatic solution to . Since and cannot be monochromatic, without loss of generality, we can assume that red and blue. Since neither of the triples , , and can be red, 4, 7, and 11 must be blue. From this, we have that is blue, contradicting our assumption.
The method to obtain the lower and upper bounds in general is similar to the one used in the above example. We first establish the lower bound.
Theorem 5. For , .
Proof. Let . To show that , it suffices to find some 2-coloring of that yields no monochromatic solution to . Using 0 and 1 as the colors, it is an easy task to check that the coloring contains no monochromatic solution to . Hence, .
Theorem 6. For , .
Proof. To show that , we must show that every 2-coloring of admits a monochromatic solution to . Using red and blue as the colors, assume, for a contradiction, that there exists a 2-coloring of with no monochromatic solution to . Since and cannot be monochromatic, without loss of generality, we can assume that red and blue. Since neither of the triples , , and can be red, , , and must be blue. From this, we have that is blue, contradicting our assumption.
By Theorems 5 and 6, we have the following corollary.
Corollary 1. For all , we have .
3. A General Lower Bound
First, suppose that . The coloring shows that . One can extend to the coloring to obtain that . Continuing in this way, we have the following.
Theorem 7. For all , .
Proof. Let and assume is an -coloring of with no monochromatic solution to . Define an -coloring of that extends as follows. For all , let , and for all , let , where (mod ). Then, contains no monochromatic solution to . So if , then and thus, . We now prove by induction that . It is clearly true for . Now, by induction hypothesis,For , the coloring shows that . We can extend this coloring to to obtain . Similarly, the coloring shows that . Continuing in this way, we can easily check that . In general, we obtain the following theorem.
Theorem 8. For all , .
Proof. Let and assume is an -coloring of with no monochromatic solution to . Define an -coloring of that extends as follows. For all , let , and for all , let , where (mod ). Then, contains no monochromatic solution to . So if , then and thus, . We now prove by induction that . It is clearly true for . Now, by induction hypothesis,
4. s2 and
In this section, we show that . The coloring shows that . We now show the other direction.
Theorem 9. We have .
Proof. To show that , we must show that every 3-coloring of admits a monochromatic solution to . Using red, blue, and green as the colors, assume, for a contradiction, that there exists a 3-coloring of with no monochromatic solution to . Since cannot be monochromatic, without loss of generality, we can assume that red and blue. Then, is red or green. We suppose red and leave the other case to the reader. Since cannot be red, then blue or green. If blue, then green, and if green, then blue or green. For a complete solution, we use the tree representation (see Figures 1 and 2).
A computer search showed us that , and so by the above upper bound, we have the following.
Theorem 10. We have .
5. Concluding Remark
In this paper, we obtain the lower bound for and find the exact value in the case of two colors. Moreover, we show that and are equal to the obtained lower bounds 43 and 94, respectively. It is interesting to prove the general formula for . It seems that .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.