In 1980, J. Beck considered the following 2-player van der Waerden game on a sequence of integers [1, N]. Player 1 and Player 2 alternately pick a previously unpicked integers in [1, N]. Player 1 wins if he has selected all members of a k-arithmetic progression. Player 2 wins if Player 1 cannot obtain all members of a k-arithmetic progression. Let W(k) be the least integer N so that the first player has a winning strategy and Beck has showed that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_875_Article_IEq1.gif" Format="GIF" Height="30" Rendition="HTML" Resolution="72" Type="Linedraw" Width="136" /> </InlineMediaObject> <EquationSource Format="TEX">$\displaystyle \lim _{k\rightarrow \infty } \left( W(k)\right) ^{1/k}=2$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <munder> <mo movablelimits="true">lim</mo> <mrow> <mi>k</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </munder> <msup> <mfenced close=")" open="("> <mi>W</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mfenced> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mn>2</mn> </mrow> </mstyle> </math></EquationSource> </InlineEquation>. In this paper, we consider the situation where Player 2 will select t integers instead of one integer in his turn and Player 1 will remain to pick only one integer in his turn. Let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_875_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$W_t(k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mi>t</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the least integer N so that the first player has a winning strategy in this setting and we show that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_875_Article_IEq3.gif" Format="GIF" Height="30" Rendition="HTML" Resolution="72" Type="Linedraw" Width="170" /> </InlineMediaObject> <EquationSource Format="TEX">$\displaystyle \lim _{k\rightarrow \infty } \left( W_t(k))\right) ^{1/k}=t+1$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <munder> <mo movablelimits="true">lim</mo> <mrow> <mi>k</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </munder> <msup> <mfenced close=")" open="("> <msub> <mi>W</mi> <mi>t</mi> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mfenced> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </mstyle> </math></EquationSource> </InlineEquation>.

On certain k-AP van der Waerden game

期刊论文
发表日期：

2025年10月14日

查看全文

Kai An Sim,
Wan Muhammad Afif Wan Ruzali,
Kok Bin Wong,
Chee Kit Ho

摘要

In 1980, J. Beck considered the following 2-player van der Waerden game on a sequence of integers [1, N]. Player 1 and Player 2 alternately pick a previously unpicked integers in [1, N]. Player 1 wins if he has selected all members of a k-arithmetic progression. Player 2 wins if Player 1 cannot obtain all members of a k-arithmetic progression. Let W(k) be the least integer N so that the first player has a winning strategy and Beck has showed that $\displaystyle \lim _{k\rightarrow \infty } \left( W(k)\right) ^{1/k}=2$ $lim_{k \to \infty} {(W (k))}^{1 / k} = 2$ . In this paper, we consider the situation where Player 2 will select t integers instead of one integer in his turn and Player 1 will remain to pick only one integer in his turn. Let $W_t(k)$ $W_{t} (k)$ be the least integer N so that the first player has a winning strategy in this setting and we show that $\displaystyle \lim _{k\rightarrow \infty } \left( W_t(k))\right) ^{1/k}=t+1$ $lim_{k \to \infty} {(W_{t}, (k)))}^{1 / k} = t + 1$ .