拉姆齐定理在埃尔德和塞克尔版本中指出$\mathrm{<trans\; data-src="2">2</trans>}$-上完备图边的着色$<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\}">\}</trans>$包含一个单色的秩序集团$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans><trans\; data-src="log">日志</trans>\mathrm{<trans\; data-src="n">n个</trans>}$在本文中，我们考虑了拉姆齐定理的两个经过充分研究的推广。改进Rödl的一个结果，我们表明存在一个常数$\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$这样，每$\mathrm{<trans\; data-src="2">2</trans>}$-上完备图边的着色$<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\}">\}</trans>$包含一个单色集团$\mathrm{<trans\; data-src="S">S公司</trans>}$其中$\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans><trans\; data-src="log">日志</trans>\mathrm{<trans\; data-src="i">我</trans>}$覆盖所有顶点$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="S">S公司</trans>}$至少是$\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="log">日志</trans><trans\; data-src="log">日志</trans><trans\; data-src="log">日志</trans>\mathrm{<trans\; data-src="n">n个</trans>}$。这与常数因子密切相关$\mathrm{<trans\; data-src="c">c（c）</trans>}$并回答了1981年Erdős的一个问题。由于模型理论中的一个问题，Väänänen问$\mathrm{<trans\; data-src="k">k</trans>}$有一个$\mathrm{<trans\; data-src="n">n个</trans>}$使得以下成立：对于每个排列$\mathrm{<trans\; data-src="\pi ">\pi </trans>}$属于$\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}$，每$\mathrm{<trans\; data-src="2">2</trans>}$-上完备图边的着色$<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\}">\}</trans>$包含一个单色集团${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="<"><</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src="<"><</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k</trans>}}$具有

$${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="\pi ">\pi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="\pi ">\pi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=">">></trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="\pi ">\pi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="\pi ">\pi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=">">></trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=">">></trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="\pi ">\pi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="k">k</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="\pi ">\pi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="k">k</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=".">.</trans>$$

也就是说，我们不仅需要单色团，而且连续顶点之间的差异必须满足规定的顺序。阿龙和埃尔德斯、哈伊纳尔和帕奇独立地肯定地回答了这个问题。阿龙进一步推测，真正的增长率应该是指数级的$\mathrm{<trans\; data-src="k">k</trans>}$.我们朝着这个猜想前进，获得了一个上界$\mathrm{<trans\; data-src="n">n个</trans>}$它是指数的幂$\mathrm{<trans\; data-src="k">k</trans>}$这改善了Shelah的一个结果，她证明了$\mathrm{<trans\; data-src="n">n个</trans>}$在$\mathrm{<trans\; data-src="k">k</trans>}$.