We study the log-rank conjecture from perspective of point-hyperplane incidence geometry. formulate following conjecture: Given a point set in $\mathbb{R}^d$ that is covered by constant-sized sets parallel hyperplanes, there exists an affine subspace accounts for large (i.e., $2^{-{\operatorname{polylog}(d)}}$) fraction incidences. Alternatively, our may be interpreted linear-algebraically as f...

We show that if α is a positive real number, n and ` are integers exceeding 1, and q is a prime power, then every simple matroid M of sufficiently large rank, with no U2,`-minor, no rank-n projective geometry minor over a larger field than GF(q), and at least αq elements, has a rank-n affine geometry restriction over GF(q). This result can be viewed as an analogue of the multidimensional densit...

We show that, for the O'Nan sporadic simple group, there is no Rwpri and (IP)2 geometry of rank 6 with a maximal parabolic subgroup isomorphic to M11 and that there is no Rwpri and (IP)2 geometry of rank 5 with a maximal parabolic subgroup isomorphic to J1 . This last result permits us to show that the Ivanov Shpectorov geometry is not Rwpri. The results obtained in this paper rely partially on...

The theory of buildings, created by J. Tits three decads ago, has ooered a uniied geometric treatment of nite simple groups of Lie type, both of classical and of exceptional type. (See Tits 19] and 20] for an exposition of that theory; also Ronan 15] and Brown 1].) Diagram geometry (see 13] for an exposition) is a generalization of the theory of buildings. It has been invented by F. Buekenhout ...