Let $P$ be a set of $2n$ points in convex position, such that $n$ points are colored red and $n$ points are colored blue. A non-crossing alternating path on $P$ of length $\ell$ is a sequence $p_1, \dots, p_\ell$ of $\ell$ points from $P$ so that (i) all points are pairwise distinct; (ii) any two consecutive points $p_i$, $p_{i+1}$ have different colors; and (iii) any two segments $p_i p_{i+1}$ and $p_j p_{j+1}$ have disjoint relative interiors, for $i \neq j$. We show that there is an absolute constant $\varepsilon > 0$, independent of $n$ and of the coloring, such that $P$ always admits a non-crossing alternating path of length at least $(1 + \varepsilon)n$. The result is obtained through a slightly stronger statement: there always exists a non-crossing bichromatic separated matching on at least $(1 + \varepsilon)n$ points of $P$. This is a properly colored matching whose segments are pairwise disjoint and intersected by common line. For both versions, this is the first improvement of the easily obtained lower bound of $n$ by an additive term linear in $n$. The best known published upper bounds are asymptotically of order $4n/3+o(n)$.

中文翻译：

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存在长交替路径

令$P$ 是一组$2n$ 点在凸面位置，使得$n$ 点为红色，$n$ 点为蓝色。长度为 $\ell$ 的 $P$ 上的非交叉交替路径是来自 $P$ 的 $\ell$ 点的序列 $p_1、\dots、p_\ell$，因此（i）所有点都是成对不同的；(ii) 任意两个连续点 $p_i$, $p_{i+1}$ 的颜色不同；(iii) 任何两个线段 $p_i p_{i+1}$ 和 $p_j p_{j+1}$ 具有不相交的相对内部，对于 $i \neq j$。我们证明存在一个绝对常数 $\varepsilon > 0$，与 $n$ 和着色无关，这样 $P$ 总是允许长度至少为 $(1 + \varepsilon)n 的非交叉交替路径$. 结果是通过一个稍微强一点的陈述获得的：在至少 $(1 + \varepsilon)n$ 个点的 $P$ 上总是存在非交叉双色分离匹配。这是一个正确着色的匹配，其段成对不相交并由公共线相交。对于这两个版本，这是通过在 $n$ 中线性的附加项对容易获得的 $n$ 下限的第一次改进。最著名的已发表上限渐近为 $4n/3+o(n)$ 阶。