Visibility cliques, cubic containers, and dense orchard cores

Abstract

The Big-Line-Big-Clique Conjecture of Kara, Por and Wood asserts that, for every fixed $k$ and $\ell$, every sufficiently large finite planar point set contains either $k$ collinear points or $\ell$ pairwise visible points. We prove a quantitative form in two structured regimes and isolate the precise ambient obstruction to the full conjecture. The main result is a deterministic cubic-container theorem. If $A\subset {\mathbb{R}}^2$ has $n$ points, no $k$ collinear points, and all but $s$ points of $A$ lie on a real cubic, then the cubic-supported part of $A$ has a visible clique cover of size $O_k(s+1)$; in particular $V(A)$ contains a clique of size ${\Omega }_k(n/(s+1))$, unless the cubic is the excluded three-line case containing only $O_k(1)$ points. Combining this with the Green-Tao structure theorem, we obtain that every $n$-point set with no $k$ collinear points and at most $Kn$ ordinary lines contains a visible clique of size ${\Omega }_{k,K}(n)$; more strongly, all but $O_K(1)$ points can be partitioned into $O_{k,K}(1)$ mutually visible sets. We also combine the cubic-container theorem with the Elekes-Szabo theorem on triple lines and cubic curves to prove the Big-Line-Big-Clique conclusion for point sets contained in any fixed irreducible algebraic curve. Finally, we prove a dense-orchard core lemma showing that the absence of a visible $K_{\ell }$ forces a positive-density subset in which every point lies on linearly many 3-rich lines, and we give a sharp one-blocker example showing why ambient blockers cannot be ignored.

18 pages, 1 figure