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A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations $E$ there is a threshold value $R_k(E)$ (the Rado number of $E$) such that for any $k$-coloring of the integers in the interval $[1,n]$, with $n \ge R_k(E)$,
Externí odkaz:
http://arxiv.org/abs/2410.21651
Autor:
Wesley, William J.
We prove new bounds for Ramsey numbers for book graphs $B_n$. In particular, we show that $R(B_{n-1},B_n) = 4n-1$ for an infinite family of $n$ using a block-circulant construction similar to Paley graphs. We obtain improved bounds for several other
Externí odkaz:
http://arxiv.org/abs/2410.03625
Autor:
Wesley, William J.
Van der Waerden's theorem states that for any positive integers $k$ and $r$, there exists a smallest value $n = w(k,r)$, called the van der Waerden number, such that every $r$-coloring of $\{1,\dots,n\}$ contains a monochromatic $k$-term arithmetic p
Externí odkaz:
http://arxiv.org/abs/2211.05167
Publikováno v:
Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation (ISSAC 2022). 2022. 333-342
Given a linear equation $\mathcal{E}$, the $k$-color Rado number $R_k(\mathcal{E})$ is the smallest integer $n$ such that every $k$-coloring of $\{1,2,3,\dots,n\}$ contains a monochromatic solution to $\mathcal E$. The degree of regularity of $\mathc
Externí odkaz:
http://arxiv.org/abs/2210.03262
Autor:
De Loera, Jesús A., Wesley, William J.
In this article we study the Ramsey numbers $R(r,s)$ through Hilbert's Nullstellensatz and Alon's Combinatorial Nullstellensatz. We give polynomial encodings whose solutions correspond to Ramsey graphs of order $n$, those that do not contain a copy o
Externí odkaz:
http://arxiv.org/abs/2209.13859
Publikováno v:
In Journal of Number Theory May 2018 186:16-34
Akademický článek
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Autor:
Wesley, William J.
Publikováno v:
Marine Corps Gazette; Aug98, Vol. 82 Issue 8, p13, 6p, 2 Diagrams