A number of important results in combinatorics, discrete geometry, and theoretical computer
science have been proved using algebraic topology. While the results are quite famous …

We discuss five fundamental results of discrete mathematics: the lemmas of Sperner and
Tucker from combinatorial topology and the theorems of Carathéodory, Helly, and Tverberg …

The well-known topological Tverberg conjecture was considered a central unsolved
problem of topological combinatorics. The conjecture asserts that for any integers $ r $, $ d …

In this paper we present a variety of problems in the interface between combinatorics and
geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these …

This survey presents an overview of the advances around Tverberg's theorem, focusing on
the last two decades. We discuss the topological, linear-algebraic, and combinatorial …

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 …

Bárány's “topological Tverberg conjecture” from 1976 states that any continuous map of an
N-simplex Δ N Δ _ N to ℝ d R^ d, for N≥(d+ 1)(r− 1), maps points from r disjoint faces in Δ N …

The" topological Tverberg conjecture" by B\'ar\'any, Shlosman and Sz\H {u} cs (1981) states
that any continuous map of a simplex of dimension $(r-1)(d+ 1) $ to $\mathbb {R}^ d $ maps …

Barycenters of polytope skeleta and counterexamples to the Topological Tverberg Conjecture,
via constraints Page 1 DOI 10.4171/JEMS/881 J. Eur. Math. Soc. 21, 2107–2116 c European …

We describe how a powerful new “constraint method” yields many different extensions of the
topological version of Tverberg's 1966 Theorem in the prime power case—and how the …