论文标题
改进了斐波那契数和其他序列的Ramsey型定理
Improved Ramsey-type theorems for Fibonacci numbers and other sequences
论文作者
论文摘要
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 progression.我们考虑了两个范德华顿数字的两种变体：$ n = n（ap_d，k; r）$的数字，是最小的值，其中每个$ r $ - 颜色为$ \ {1，\ dots，n \} $都包含单色$ k $ k $ k $ -k $ -k $ - term arithmetic arithmetic atith atith arith arithmet ant term在$ d $ $ d $ $ d $ $ d $ $ n = n = n =Δ（d $ n =Δ（d）中， $ r $ -Coloring $ \ {1，\ dots，n \} $包含一个序列$ x_1 <\ dots <x_k $，其中连续项之间的差异为$ d $的成员。当$ d $设置为fibonacci数字$ f $并为最大$ r $提供改进的界限时，我们研究了案例。此外，我们为其他集合$ d $提供了一些$δ（d，k; r）$的计算数据。
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 progression. We consider two variants of van der Waerden numbers: the numbers $n = n(AP_D,k;r)$, the smallest value where every $r$-coloring of $\{1,\dots,n\}$ contains a monochromatic $k$-term arithmetic progression with common difference in $D$, and the numbers $n = Δ(D,k;r)$, the smallest value $n$ where every $r$-coloring of $\{1,\dots,n\}$ contains a sequence $x_1 < \dots < x_k$ where the differences between consecutive terms are members of $D$. We study the case when $D$ is set of Fibonacci numbers $F$ and give improved bounds for the largest $r$ where $n(AP_F,k;r)$ and $Δ(F,k;r)$ exist for all $k$. Moreover, we give some computational data on $Δ(D,k;r)$ for other sets $D$.
