. Let E m denote a system of equations/inequalities in m variables. Generalizing the concept of a Schur number, the Rado number R t ( E m ) is deﬁned to be the least natural number such that every t -coloring of the elements of the set { 1 , 2 , . . . , R t ( E m ) } contains a monochromatic solution to E m . In this paper, we review the relevant background for R t ( E m ) and consider two variations. First, we deﬁne the weakened Rado number, in which solutions to E m are sought that use at most s < t colors. We compute a few special cases, then turn our attention to the rainbow Rado number RR ( E m , n ), deﬁned to be the minimum number of colors such that every coloring of { 1 , 2 , . . . , n } contains a rainbow solution to E m (a solution in which all m variables receive distinct colors). We evaluate rainbow Rado numbers for two speciﬁc systems of inequalities, then conclude with some directions for future work on this topic. 2010

Some variations of multicolor rado numbers

Extremal Combinatorics is one of the central and heavily contributed areas in discrete mathematics,
and has seen an outstanding growth during the last few decades. In general, it deals
with problems regarding determination and/or estimation of the maximum or the minimum size
of a combinatorial structure that satisfies a certain combinatorial property. Problems in Extremal
Combinatorics are often related to theoretical computer science, number theory, geometry, and information
theory. In this thesis, we work on some well-known problems (and on their variants) in
Extremal Combinatorics concerning the set of integers as the combinatorial structure.

The van der Waerden number w(k;t_0,t_1,...,t_{k-1}) is the smallest positive
integer n such that every k-colouring of 1, 2, . . . , n contains a monochromatic
arithmetic progression of length t_j for some colour j in {0,1,...,k-1}. We have
determined five new exact values with k=2 and conjectured several van der Waerden numbers
of the form w(2;s,t), based on which we have formulated a polynomial
upper-bound-conjecture of w(2; s, t) with fixed s. We have provided an efficient SAT
encoding for van der Waerden numbers with k>=3 and computed three new van der Waerden
numbers using that encoding. We have also devised an efficient problem-specific
backtracking algorithm and computed twenty-five new van der Waerden numbers with k>=3
using that algorithm.

We have proven some counting properties of arithmetic progressions and some unimodality
properties of sequences regarding arithmetic progressions. We have generalized Szekeres’
conjecture on the size of the largest sub-sequence of 1, 2, . . . , n without an
arithmetic progression of length k for specific k and n; and provided a construction for
the lower bound corresponding to the generalized conjecture.

A Strict Schur number S(h,k) is the smallest positive integer n such that every
2-colouring of 1,2,...,n has either a blue solution to x_1 +x_2 +···+x_{h-1} = x_h
where x_1 < x_2 < ··· < x_h, or a red solution to x_1+x_2+···+x_{k-1} =x_k where
x_1 <x_2 <···<x_k. We have proven the exact formula for S(3, k).

https://spectrum.library.concordia.ca/976929/1/Ahmed_PhD_S2103.pdf

Some Results in Extremal Combinatorics

Ramsey Theory on the Integers: Second Edition

Let represent the equation x1 + x2 + ⋅⋅⋅ + xt − 1 = xt. For k ≥ 1, 0 ≤ i ≤ k − 1, and ti ≥ 3, the generalized Schur number S(k; t0, t1, …, tk − 1) is the least positive integer m such that for every k-coloring of {1, 2, …, m}, there exists an i ∈ {0, 1, …, k − 1} such that there exists a solution to that is monochromatic in color i. In this article, we report 26 previously unknown values of S(k; t0, t1, …, tk − 1) and conjecture that for 4 ≤ t0 ≤ t1 ≤ t2, S(3; t0, t1, t2) = t2t1t0 − t2t1 − t2 − 1.

On Generalized Schur Numbers

In this study, we focus on the concept of the 2-color off-diagonal generalized weak Schur numbers, denoted as WS(2; k1, k2). These numbers are defined for integers ki ≥ 2, where i = 1, 2, as the smallest integer M, such that any 2-coloring of the integer interval [1, M] must contain a 2-colored solution to the equation Ekj: x1 + x2 + ... + xkj = xkj+1 for j = 1,2, with the condition that xi ≠ xj when i ≠ j. Our objective is to determine lower bounds for these 2-color off-diagonal generalized weak Schur numbers and demonstrate that in several cases, these lower bounds match the exact values. 

Lower bounds and exact values of the 2-color off-diagonal generalized weak Schur numbers WS(2;k,k) (Brief Announcement)

Strict Schur Numbers.

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Strict Schur Numbers.

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Citations

Some variations of multicolor rado numbers

Some Results in Extremal Combinatorics

Ramsey Theory on the Integers: Second Edition

On Generalized Schur Numbers

Lower bounds and exact values of the 2-color off-diagonal generalized weak Schur numbers WS(2;k,k) (Brief Announcement)

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