Let N=W(2, t^2+1), where W(r, t) is the van der Waerden's function, and let χbe a coloring of {1

Step 1: Understand the problem and definitions.u000AWe are given a function $N u003D W(2, t^2 + 1)$, whe

Question

Let $N=W\left(2, t^2+1\right)$, where $W(r, t)$ is the van der Waerden's function, and let $\chi$ be a coloring of $\{1, \ldots, N\}$ in two colors. Show that there exists a $t$-term arithmetic progression $\{a+i \cdot d: i=0,1, \ldots, t-1\}$ which together with its difference $d$ is monochromatic, i.e., $\chi(d)=\chi(a+i \cdot d)$ for every $i<t$. Hint: Van der Waerden's theorem gives a monochromatic arithmetical progression $\left\{a+j \cdot d: j \leq t^2\right\}$ with $t^2$ terms. Then either some $j \cdot d$, with $1 \leq j \leq t$, gets the same color or all the numbers $d, 2 d, \ldots, t d$ get the opposite color.

   Let $N=W\left(2, t^2+1\right)$, where $W(r, t)$ is the van der Waerden's function, and let $\chi$ be a coloring of $\{1, \ldots, N\}$ in two colors. Show that there exists a $t$-term arithmetic progression $\{a+i \cdot d: i=0,1, \ldots, t-1\}$ which together with its difference $d$ is monochromatic, i.e., $\chi(d)=\chi(a+i \cdot d)$ for every $i<t$. Hint: Van der Waerden's theorem gives a monochromatic arithmetical progression $\left\{a+j \cdot d: j \leq t^2\right\}$ with $t^2$ terms. Then either some $j \cdot d$, with $1 \leq j \leq t$, gets the same color or all the numbers $d, 2 d, \ldots, t d$ get the opposite color.

Extremal Combinatorics: With Applications in Computer Science

Stasys Jukna 2nd Edition

Chapter 26, Problem 2

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We are given a function $N = W(2, t^2 + 1)$, where $W(r, t)$ is the van der Waerden's function. This function guarantees that for any coloring of $\{1, \ldots, N\}$ in $r$ colors, there exists a monochromatic arithmetic progression of length $t$. In our case, $r = Show more…

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Let $N=W\left(2, t^2+1\right)$, where $W(r, t)$ is the van der Waerden's function, and let $\chi$ be a coloring of $\{1, \ldots, N\}$ in two colors. Show that there exists a $t$-term arithmetic progression $\{a+i \cdot d: i=0,1, \ldots, t-1\}$ which together with its difference $d$ is monochromatic, i.e., $\chi(d)=\chi(a+i \cdot d)$ for every $i<t$. Hint: Van der Waerden's theorem gives a monochromatic arithmetical progression $\left\{a+j \cdot d: j \leq t^2\right\}$ with $t^2$ terms. Then either some $j \cdot d$, with $1 \leq j \leq t$, gets the same color or all the numbers $d, 2 d, \ldots, t d$ get the opposite color.

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