Disjointness graphs of segments in the space

The disjointness graph G = G(𝒮) of a set of segments 𝒮 in ${\mathbb{R}^d}$ , $$d \ge 2$$ , is a graph whose vertex set is 𝒮 and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies $\chi (G) \le {(\omega (G))^4} + {(\omega (G))^3}$ , where ω(G) denotes the clique number of G. It follows that 𝒮 has Ω(n1/5) pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments. We show that computing ω(G) and χ(G) for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colourings of G in which the number of colours satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free (ω(G) = 2), but whose chromatic numbers are arbitrarily large.

PEMBELAJARAN BERBASIS DIGITAL: STUDI PENGGUNAAN GEOGEBRA BERBANTUAN E-LEARNING UNTUK MENINGKATKAN HASIL BELAJAR MATEMATIKA

Abstrak Penelitian ini menggambarkan pembelajaran berbasis digital, yaitu penggunaan Geogebra berbantuan e-learning, untuk meningkatkan hasil belajar mahasiswa pada konsep bidang dan garis dalam ruang. Penelitian ini adalah pre-experimental dengan jenis one-shot case study dengan partisipan mahasiswa calon dosen matematika pada semester II yang terdiri dari 14 orang laki-laki dan 30 orang perempuan dari salah satu universitas di Indonesia. Hasil penelitian menunjukkan penggunaan Geogebra berbantuan e-learning efektif untuk meningkatkan hasil belajar mahasiswa pada konsep bidang dan garis dalam ruang. Hal ini terlihat dari sebagian besar mahasiswa memperoleh hasil belajar yang maksimal, dan hanya sebagian kecil mahasiswa yang hasil belajarnya masih minimal. Penelitian ini merekomendasikan penggunaan Geogebra dan e-learning pada pembelajaran konsep geometri lainnya. AbstractThis study describes digital-based learning, namely the use of Geogebra assisted by e-learning, to improve student learning outcomes in the concepts of fields and lines in space. This study was pre-experimental with a type of one-shot case study with participants of mathematics teacher candidates in the second semester consisting of 14 men and 30 women from one university in Indonesia. The results showed that the use of Geogebra assisted by e-learning was effective in improving student learning outcomes in the concepts of fields and lines in space. This can be seen from the majority of students obtaining maximum learning outcomes, and only a small percentage of students whose learning outcomes are still minimal. This study recommends the use of Geogebra and e-learning in learning other geometry concepts.

The Completeness of the Real Line

It is widely taken for granted that physical lines are real lines, i.e., that the arithmetical structure of the real numbers uniquely matches the geometrical structure of lines in space; and that other number systems, like Robinson’s hyperreals, accordingly fail to fit the structure of space. Intuitive justifications for the consensus view are considered and rejected. Insofar as it is justified at all, the conviction that physical lines are real lines is a scientific hypothesis which we may one day reject.