A family of vectors in [k]n is said to be intersecting if any two of its elements agree on at least one coordinate. We prove, for fixed k ≥ 3, that the size of any intersecting subfamily of [k]n invariant under a transitive group of symmetries is o(kn), which is in stark contrast to the case of the Boolean hypercube (where k = 2). Our main contribution addresses limitations of existing technology: while there are now methods, first appearing in work of Ellis and the third author, for using spectral machinery to tackle problems in extremal set theory involving symmetry, this machinery relies crucially on the interplay between up-sets, biased product measures, and threshold behaviour in the Boolean hypercube, features that are notably absent in the problem considered here. To circumvent these barriers, introducing ideas that seem of independent interest, we develop a variant of the sharp threshold machinery that applies at the level of products of posets.

We construct a ring homomorphism comparing the tautological ring, fixing a point, of a closed smooth manifold with that of its stabilisation by S2a×S2b.

Let X be a topologically stratified space, p be any perversity on X and k be a field. We show that the category of p-perverse sheaves on X, constructible with respect to the stratification and with coefficients in k, is equivalent to the category of finite-dimensional modules over a finite-dimensional algebra if and only if X has finitely many strata and the same holds for the category of local systems on each of these. The main component in the proof is a construction of projective covers for simple perverse sheaves.

We prove that the special-value conjecture for the zeta function of a proper, regular, flat arithmetic surface formulated in [6] at $s=1$ is equivalent to the Birch and Swinnerton-Dyer conjecture for the Jacobian of the generic fibre. There are two key results in the proof. The first is the triviality of the correction factor of [6, Conjecture 5.12], which we show for arbitrary regular proper arithmetic schemes. In the proof we need to develop some results for the eh-topology on schemes over finite fields which might be of independent interest. The second result is a different proof of a formula due to Geisser, relating the cardinalities of the Brauer and the Tate–Shafarevich group, which applies to arbitrary rather than only totally imaginary base fields.

We construct a calculus of functors in the spirit of orthogonal calculus, which is designed to study ‘functors with reality’ such as the Real classifying space functor, . The calculus produces a Taylor tower, the n-th layer of which is classified by a spectrum with an action of . We further give model categorical considerations, producing a zigzag of Quillen equivalences between spectra with an action of and a model structure on the category of input functors which captures the homotopy theory of the n-th layer of the Taylor tower.

It has been conjectured that, for any fixed \[{\text{r}} \geqslant 2\] and sufficiently large n, there is a monochromatic Hamiltonian Berge-cycle in every \[({\text{r}} - 1)\]-colouring of the edges of \[{\text{K}}_{\text{n}}^{\text{r}}\], the complete r-uniform hypergraph on n vertices. In this paper we prove this conjecture.

For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated with a triangle functor from the category on the right to the category on the left. For a morphic enhancement of a triangulated category $\mathcal {T}$, there are three explicit ideals of the enhancing category, whose corresponding factor categories are all equivalent to the module category over $\mathcal {T}$. Examples related to inflation categories and weighted projective lines are discussed.