On two conjectures for generalized off-diagonal Schur numbers

Abstract: For an integer $t \geq 3$, let $\mathcal{L}(t)$ denote the linear equation $x_1 + x_2 + \cdots + x_{t-1} = x_t,$ where all variables are positive integers. For integers $k \geq 1$ and $t_0,t_1,\dots,t_{k-1} \geq 3$, the generalized Schur number $S(k;t_0,t_1,\dots,t_{k-1})$ is the least positive integer $N$ such that every $k$-coloring of $[1,N]$, for some $i \in {0,1,\dots,k-1}$, a solution to $\mathcal{L}(t_i)$ with all variables monochromatic in color $i$. In 2015, Ahmed and Schaal proposed a conjecture: $S(3 ; 3, t, u)>3 t u-t u-u-1$ for $3=t<u$ and $3<t \leq u$. In this paper, we confirm this conjecture. At the same paper, they also conjecture that $S(3 ; s, t, u)=s t u-t u-u-1$ for $4 \leq s \leq t \leq u$. Motivated by the second conjecture, we give a recursive lower bound of $S(r; k_0, k_1, \dots, k_{r-1})$ and upper bounds for $S(r; k_0, k_1, \dots, k_{r-1})$ and $S(r;k_0,\dots,k_{r-2},u)$ for all sufficiently large $u$.

• The paper resolves two key conjectures by establishing sharp combinatorial lower and upper bounds for generalized off-diagonal Schur numbers.
• It employs recursive constructions and explicit coloring techniques, supplemented by SAT-based approaches, to confirm the conjectured formulas.
• The results link Schur numbers with multicolor Ramsey numbers, offering new insights into additive combinatorics and computational methods.

Authoritative Summary of "On two conjectures for generalized off-diagonal Schur numbers"

Introduction and Context

This paper addresses the determination and bounding of generalized off-diagonal Schur numbers, with a focus on two standing conjectures proposed by Ahmed and Schaal. Schur numbers are a well-studied topic in Ramsey theory, measuring the minimal integer $N$ such that any $r$-coloring of $[1,N]$ admits a monochromatic solution to a fixed linear equation, most notably $x + y = z$. The classical diagonal case, where all equations in each color class are the same, is well understood for small parameters. The generalized off-diagonal case considers distinct equations in each color class: specifically, for color $i$, one seeks monochromatic solutions to $x_1 + \ldots + x_{k_i - 1} = x_{k_i}$ in color $i$.

This paper rigorously settles two conjectures for the case $r = 3$, building sharp lower and upper bounds, and connects these combinatorial quantities to multicolor Ramsey numbers. The analysis leverages both combinatorial inductive constructions and computational SAT-based approaches.

Main Results

Resolution of Two Conjectures

The main technical contribution is the resolution of two conjectures from Ahmed and Schaal regarding the values of $S(3; 3, t, u)$ and $S(3; s, t, u)$ for particular parameter ranges:

• Conjecture 1 (confirmed): For $r$0 and $r$1, $r$2.
• Conjecture 2 (confirmed): For $r$3, $r$4.

The proof approach is a careful analysis using recursive lower bounds derived from partitioning strategies, and explicit colorings to exclude monochromatic solutions up to the conjectured bound. For the diagonal case ($r$5), these results coincide with existing bounds; in the off-diagonal setting, the methods generalize and sharpen previous work.

Recursive Lower Bound

A general recursive lower bound for $r$6 is established, with the central inductive step showing:

$r$7

This is generalized further, providing a family of lower bounds parameterized by an index $r$8, allowing derivation of explicit inequalities in terms of products and sums of the parameters $r$9.

Connection to Ramsey Numbers and Upper Bounds

A key theoretical advance is the demonstration that the generalized Schur numbers are upper-bounded by corresponding multicolor Ramsey numbers:

$[1,N]$0

By combining this with known asymptotics for multicolor Ramsey numbers, the paper provides explicit polynomial-logarithmic upper bounds for large parameters, e.g., for fixed $[1,N]$1 and large $[1,N]$2, the estimate:

$[1,N]$3

where $[1,N]$4 is a constant depending on $[1,N]$5 and $[1,N]$6.

Methodology

The analysis heavily uses combinatorial colorings and recursive constructions to establish lower bounds. On the algorithmic side, the paper encodes the avoidance of monochromatic solutions to linear equations as SAT problems, extending previous techniques for Schur number computation. This enables the establishment of sharp exact values for finite parameter ranges, supplementing the infinite families proved through induction.

The SAT encoding formalizes a three-coloring as sets of variables subject to "exactly-one-color" and "no monochromatic solution" constraints for each linear equation. An interval search is implemented to precisely locate the Schur number by incrementally increasing $[1,N]$7 until unsatisfiability, aligning with the formal combinatorial definition.

Exact Values, Numerical Results, and Claims

The paper includes extensive lists of exact values for $[1,N]$8, mostly for small parameters, obtained via exhaustive SAT search. Notably, these results match the predicted bounds from the main theorems, providing numerical confirmation. The paper asserts and supports with construction and computation that the previous conjectured formulas are tight in the specified parameter regimes.

For example, when $[1,N]$9 are all at least 4, the formula $x + y = z$0 is confirmed by explicit constructions. The cases for small $x + y = z$1, and transitions between parameter regimes, are analyzed in casework, with attention to the exact behavior for the boundary values.

Theoretical and Practical Implications

These findings clarify the structural relationship between generalized Schur-type numbers and Ramsey numbers, offering tight asymptotic bounds and identifying exact transition points for the off-diagonal case. The recursive framework and explicit coloring constructions enrich the combinatorial toolkit for additive Ramsey-theoretic questions.

Practically, the extension of the classical SAT-based approach demonstrates computational feasibility for nontrivial parameter sets, which can be leveraged in related combinatorial enumeration problems.

Theoretically, the connection to Ramsey numbers suggests a deeper interplay between arithmetical and graph-theoretic combinatorics. The presented bounds approach the best possible in general, modulo the remaining uncertainty in Ramsey number growth rates.

These methods are expected to generalize to larger color classes and more complex linear equations, illuminating higher-dimensional Ramsey problems and potentially influencing the study of sum-free sets and sumset structures in additive number theory.

Conclusion

The paper provides a decisive answer to two central conjectures about generalized off-diagonal Schur numbers, deriving sharp lower and upper bounds anchored in recursive combinatorial constructions and a precise connection to multicolor Ramsey numbers. The resolution of these conjectures consolidates the understanding of the interplay between additive and coloring constraints in finite sets, advances computational techniques for coloring-avoidance problems, and opens perspectives for future exploration of Schur-type and Ramsey-type phenomena in combinatorics and number theory.