The Grünbaum-Hadwiger-Ramos hyperplane mass partition problem was introduced by Grünbaum (1960) in a special case and in general form by Ramos (1996). It asks for the "admissible" triples (d, j, k) such that for any j masses in R d there are k hyperplanes that cut each of the masses into 2 k equal parts. Ramos' conjecture is that the Avis-Ramos necessary lower bound condition dk ≥ j(2 k − 1) is also sufficient.We develop a "join scheme" for this problem, such that non-existence of anthat extends a test map on the subspace of (S d ) * k where the hyperoctahedral group S ± k acts non-freely, implies that (d, j, k) is admissible.For the sphere (S d ) * k we obtain a very efficient regular cell decomposition, whose cells get a combinatorial interpretation with respect to measures on a modified moment curve. This allows us to apply relative equivariant obstruction theory successfully, even in the case when the difference of dimensions of the spheres (S d ) * k and S(W k ⊕ U ⊕j k ) is greater than one. The evaluation of obstruction classes leads to counting problems for concatenated Gray codes.Thus we give a rigorous, unified treatment of the previously announced cases of the Grünbaum-Hadwiger-Ramos problem, as well as a number of new cases for Ramos' conjecture.

Many of the strengthenings and extensions of the topological Tverberg theorem can be derived with surprising ease directly from the original theorem: For this, we introduce a proof technique that combines a concept of ‘Tverberg unavoidable subcomplexes’ with the observation that Tverberg points that equalize the distance from such a subcomplex can be obtained from maps to an extended target space. Thus, we obtain simple proofs for many variants of the topological Tverberg theorem, such as the colored Tverberg theorem of Živaljević and Vrećica (‘The colored Tverberg's problem and complexes of injective functions’, J. Combin. Theory, Ser. A 61 (1992) 309–318). We also get a new strengthened version of the generalized van Kampen–Flores theorem by Sarkaria (‘A generalized van Kampen–Flores theorem’, Proc. Amer. Math. Soc. 11 (1991) 559–565) and Volovikov (‘On the van Kampen–Flores theorem’, Math. Notes (5) 59 (1996) 477–481), an affine version of their ‘j‐wise disjoint’ Tverberg theorem, and a topological version of Soberón's result (‘Equal coefficients and tolerance in coloured Tverberg partitions’, Proceeding of the 29th Annual Symposium on Computational Geometry (SoCG) (ACM, Rio de Janeiro, 2013), 91–96) on Tverberg points with equal barycentric coordinates.

We prove a "Tverberg type" multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.

Bárány's "topological Tverberg conjecture" from 1976 states that any continuous map of an N -simplex ∆ N to R d , for N ≥ (d + 1)(r − 1), maps points from r disjoint faces in ∆ N to the same point in R d . The proof of this result for the case when r is a prime, as well as some colored version of the same result, using the results of Borsuk-Ulam and Dold on the non-existence of equivariant maps between spaces with a free group action, were main topics of Matoušek's 2003 book "Using the Borsuk-Ulam theorem."In this paper we show how advanced equivariant topology methods allow one to go beyond the prime case of the topological Tverberg conjecture.First we explain in detail how equivariant cohomology tools (employing the Borel construction, comparison of Serre spectral sequences, Fadell-Husseini index, etc.) can be used to prove the topological Tverberg conjecture whenever r is a prime power. Our presentation includes a number of improved proofs as well as new results, such as a complete determination of the Fadell-Husseini index of chessboard complexes in the prime case.Then we introduce the "constraint method," which applied to suitable "unavoidable complexes" yields a great variety of variations and corollaries to the topological Tverberg theorem, such as the "colored" and the "dimension-restricted" (Van Kampen-Flores type) versions.Both parts have provided crucial components to the recent spectacular counter-examples in high dimensions for the case when r is not a prime power.