Dense Graphs Have Rigid Parts

While the problem of determining whether an embedding a graph G in $${\mathbb {R}}^2$$ is infinitesimally rigid well understood, specifying given or not still hard task that usually requires ad hoc arguments. In this paper, we show every (not necessarily generic) dense enough (concretely, with at least $$C_0n^{3/2}(\log n)^{\beta }$$ edges, for some absolute constants $$C_0>0$$ and $$\beta $$ ), which satisfies very mild general position requirements (no three vertices are embedded to common line), must have subframework size rigid. For proof use connection, established Raz (Discrete Comput. Geom. 58(4), 986–1009 (2017)), between notion rigidity configurations lines {R}}^3$$ . This connection allows us properties line Guth Katz (Ann. Math. 181(1), 155–190 (2015)). fact, our extended version result; extension need proved by János Kollár appendix paper. We do know assumption on number edges being $$\Omega (n^{3/2}\log n)$$ tight, provide construction shows requiring (n\log necessary.