Abstract An eight-partition of a finite set of points (respectively, of a continuous mass distribution) in $$\mathbb {R}^3$$ R 3 consists of three planes that divide the space into 8 octants, such that each open octant contains at most 1/8 of the points (respectively, of the mass). In 1966, Hadwiger showed that any mass distribution in $$\mathbb {R}^3$$ R 3 admits an eight-partition; moreover, one can prescribe the normal direction of one of the three planes. The analogous result for finite point sets follows by a standard limit argument. We prove the following variant of this result: any mass distribution (or point set) in $$\mathbb {R}^3$$ R 3 admits an eight-partition for which the intersection of two of the planes is a line with a prescribed direction. Moreover, we present an efficient algorithm for calculating an eight-partition of a set of n points in $$\mathbb {R}^3$$ R 3 (with prescribed normal direction of one of the planes) in time $$O (n^{7/3})$$ O ( n 7 / 3 ) . A preliminary version of this work appeared in SoCG’24 (Aronov et al., 40th International Symposium on Computational Geometry, 2024).

We study the complexity of a class of promise graph homomorphism problems. For a fixed graph H, the H-colouring problem is to decide whether a given graph has a homomorphism to H. By a result of Hell and Nešetřil, this problem is NP-hard for any non-bipartite loop-less graph H. Brakensiek and Guruswami [SODA 2018] conjectured the hardness extends to promise graph homomorphism problems as follows: fix a pair of non-bipartite loop-less graphs G, H such that there is a homomorphism from G to H, it is NP-hard to distinguish between graphs that are G-colourable and those that are not H-colourable. We confirm this conjecture in the cases when both G and H are 4-colourable. This is a common generalisation of previous results of Khanna, Linial, and Safra [Comb. 20(3): 393-415 (2000)] and of Krokhin and Opršal [FOCS 2019]. The result is obtained by combining the algebraic approach to promise constraint satisfaction with methods of topological combinatorics and equivariant obstruction theory.

An eight-partition of a finite set of points (respectively, of a continuous mass distribution) in ℝ³ consists of three planes that divide the space into 8 octants, such that each open octant contains at most 1/8 of the points (respectively, of the mass). In 1966, Hadwiger showed that any mass distribution in ℝ³ admits an eight-partition; moreover, one can prescribe the normal direction of one of the three planes. The analogous result for finite point sets follows by a standard limit argument. We prove the following variant of this result: Any mass distribution (or point set) in ℝ³ admits an eight-partition for which the intersection of two of the planes is a line with a prescribed direction. Moreover, we present an efficient algorithm for calculating an eight-partition of a set of n points in ℝ³ (with prescribed normal direction of one of the planes) in time O^*(n^{5/2}).

A linearly ordered (LO) $k$-colouring of a hypergraph is a colouring of its vertices with colours $1, \dots, k$ such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO $k$-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the `linearly ordered chromatic number' of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO $3$-colourable, and the case that it is not even LO $4$-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opršal, Wrochna, and Živný (2023).