New Proofs for the Disjunctive Rado Number of the Equations $$x_1-x_2=a$$ and $$x_1-x_2=b$$

Dileep, A.; Moondra, Jai; Tripathi, A. N.

doi:10.1007/s00373-021-02400-y

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“…Johnson & Schaal gave necessary and sufficient conditions for the existence of the 2-color disjunctive Rado number for the equations x 1 − x 2 = a and x 1 − x 2 = b for all pairs of distinct positive integers a, b, and also determined exact values when it exists. The present authors provided alternate proofs for the same result; see [2]. Johnson & Schaal [8] also determined exact values for the pair of equations ax 1 = x 2 and bx 1 = x 2 whenever a, b are distinct positive integers.…”

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confidence: 74%

“…. , y = x + sa k } for any s ∈ N. As warmup to our results for A , we use Theorems 7 and 11 to give an alternate proof of Johnson & Schaal's result for R d (A ) in Theorem 12; this proof was previously observed in [2]. We determine conditions for the existence of R d (A ) in Theorem 13, establish general upper and lower bounds on R d (A ) in Theorems 15 and 16 respectively, and determine R d (A ) for large enough a k in Theorem 20.…”

Section: A General Theorem For Equations Of Two Variablesmentioning

confidence: 99%

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On Disjunctive Rado Numbers for Some Sets of Equations

Dileep,

Moondra,

Tripathi

2024

Electron. J. Combin.

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Given a set of linear equations $S$ with positive integral parameters $a_1,\ldots,a_k$, $k \ge 2$, the disjunctive Rado number for the set $S$ is the least positive integer $R=R_d(S)$, if it exists, such that every $2$-coloring $\chi$ of the integers in $[1, R]$ admits a monochromatic solution to at least one equation in $S$. We give conditions for the existence of $R_d(S)$, and also give general upper and lower bounds on $R_d(S)$, when $S$ is a set of additive equations $\{y=x+a_1,\ldots,y=x+a_k\}$. We also determine $R_d(S)$ when $\max a_i$ is large enough, or when $a_1,\ldots,a_k$ form an arithmetic or geometric progression. We also give conditions for the existence of $R_d(S)$ when $S$ is a set of multiplicative equations $\{y=a_1 x,\ldots,y=a_k x\}$. Further, we give a general search-based algorithm to determine $R_d(S)$ when $S$ is a system of equations in two variables, given an upper bound on $R_d(S)$ and an algorithm to determine solutions to $S$. This algorithm runs in time $O(ka_k \log a_k)$ for the case of additive equations, which is exponentially better than the brute-force algorithm for the problem.

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confidence: 74%

Section: A General Theorem For Equations Of Two Variablesmentioning

confidence: 99%