Türanordnungen

Türanordnungen is a term that may appear in German-language discussions of Turán-type problems in extremal graph theory. The phrase is not standard across mathematics, where the more common references are Turán graphs, Turán’s theorem, and Turán-type constructions. When Türanordnungen is used, it usually denotes a particular arrangement or partition of a graph’s vertex set that realizes or approximates the extremal configuration described by Turán-type results.

In the typical Turán setting, one seeks the maximum number of edges in a graph on n vertices that avoids a given subgraph H (often a complete graph K_r). The extremal construction achieving this bound is the Turán graph T(n, r−1): the vertex set is partitioned into r−1 parts of as equal size as possible, with all edges between different parts and none within a part. A Türanordnung, then, would be an arrangement of vertices into such parts, or more generally a partition that mirrors the structure that extremal graphs aim to approximate.

Historically, the concept originates from Pál Turán, who established fundamental results in graph theory in the 1940s. In German texts, you may encounter variants such as Türanordnung, Turánordnung, or Turán-Partition; nevertheless, these are less standard than the widely used terminology Turán graph and Turán’s theorem. Türanordnungen are primarily of interest in proofs and constructions within extremal combinatorics, where the way vertices are grouped can influence edge counts, stability results, and asymptotic behavior.

See also: Turán’s theorem, Turán graph, extremal graph theory.