Point-hyperplane incidence geometry and the log-rank conjecture

We study the log-rank conjecture from the perspective of point-hyperplane incidence geometry. We formulate the following conjecture: Given a point set in R that is covered by constant-sized sets of parallel hyperplanes, there exists an affine subspace that accounts for a large (i.e., 2) fraction of the incidences, in the sense of containing a large fraction of the points and being contained in a large fraction of the hyperplanes. (In other words, the point-hyperplane incidence graph for such configurations has a large complete bipartite subgraph.) Alternatively, our conjecture may be interpreted linear-algebraically as follows: Any rank-d matrix containing at most O(1) distinct entries in each column contains a submatrix of fractional size 2, in which each column contains one distinct entry. We prove that our conjecture is equivalent to the log-rank conjecture; the crucial ingredient of this proof is a reduction from bounds for parallel k-partitions to bounds for parallel (k−1)-partitions. We also introduce an (apparent) strengthening of the conjecture, which relaxes the requirements that the sets of hyperplanes be parallel. Motivated by the connections above, we revisit well-studied questions in point-hyperplane incidence geometry without structural assumptions (i.e., the existence of partitions), and in particular, we initiate the study of complete bipartite subgraph size in incidence graphs in the regime where dimension is not a constant. We give an elementary argument for the existence of complete bipartite subgraphs of density Ω(ǫ/d) in any d-dimensional configuration with incidence density ǫ (qualitatively matching previous results, which had implicit dimension-dependent constants and required sophisticated incidencegeometric techniques). We also improve an upper-bound construction of Apfelbaum and Sharir [AS07], yielding a configuration whose complete bipartite subgraphs are exponentially small and whose incidence density is Ω(1/ √ d). Finally, we discuss various constructions (due to others) of products of Boolean matrices which yield configurations with incidence density Ω(1) and bipartite subgraph density 2 √ , and pose several questions for this special case in the alternative language of extremal set combinatorics. Our framework and results may help shed light on the difficulty of improving Lovett’s Õ( √ rank(f)) bound [Lov16] for the log-rank conjecture; in particular, any improvement on this bound would imply the first bipartite subgraph size bounds for parallel 3-partitioned configurations which beat our generic bounds for unstructured configurations. Harvard College. noahsinger@college.harvard.edu. Work supported by the Herchel Smith Fellowship. School of Engineering and Applied Sciences, Harvard University. madhu@cs.harvard.edu. Supported in part by a Simons Investigator Award and NSF Award CCF 1715187.