AI 中文总结
研究哈博思猜想及克莱伯加强猜想，将克莱伯猜想归结为至多五个顶点特殊多边形的局部有理距离陈述，证明非退化四边形在所有情况下内部整数对角线上存在有理距离点。
AI 中文摘要
哈博思猜想指出每个平面图都有一个无交叉直线绘制，其中每条边具有整数长度。克莱伯的加强要求顶点本身具有整数坐标。在本系列论文中，我们朝着解决这些猜想取得进展。我们将克莱伯猜想简化为至多五个顶点的特殊多边形的局部有理距离陈述。三角形情况已知。本文中，我们证明了非退化四边形在所有情况下内部整数对角线上存在有理距离点。在后续论文中，我们将专注于非退化五边形，然后是退化四边形和退化五边形。
英文摘要
Harborth's conjecture states that every planar graph has a crossing-free straight-line drawing in which every edge has an integer length. Kleber's strengthening asks for the vertices themselves to have integer coordinates. In this series of papers, we make progress towards settling these conjectures. We reduce Kleber's conjecture to local rational-distance statements for special polygons with at most five vertices. The triangle case is known from the results of Almering and Berry. In this paper, we prove the existence of a rational-distance point on an interior integer diagonal in all the cases for non-degenerate quadrilaterals. In the upcoming papers, we focus on non-degenerate pentagons and then degenerate quadrilaterals and degenerate pentagons.
