## Section: New Results

### Discrete Geometric structures

Participants : Xavier Goaoc, Galatée Hemery Vaglica.

#### Shatter functions with polynomial growth rates

We study how a single value of the shatter function of a set system restricts its asymptotic growth. Along the way, we refute a conjecture of Bondy and Hajnal which generalizes Sauer's Lemma. [12]

#### The discrete yet ubiquitous theorems of Caratheodory, Helly, Sperner, Tucker, and Tverberg

We discuss five discrete results: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carathéodory, Helly, and Tverberg from combinatorial geometry. We explore their connections and emphasize their broad impact in application areas such as game theory, graph theory, mathematical optimization, computational geometry, etc. [13]

#### Shellability is NP-complete

We prove that for every $d\ge 2$, deciding if a pure, $d$-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every $d\ge 2$ and $k\ge 0$, deciding if a pure, $d$-dimensional, simplicial complex is $k$-decomposable is NP-hard. For $d\ge 3$, both problems remain NP-hard when restricted to contractible pure $d$-dimensional complexes. Another simple corollary of our result is that it is NP-hard to decide whether a given poset is CL-shellable. [15]

#### An Experimental Study of Forbidden Patterns in Geometric Permutations by Combinatorial Lifting

We study the problem of deciding if a given triple of permutations can be realized as geometric permutations of disjoint convex sets in ${\mathbb{R}}^{3}$. We show that this question, which is equivalent to deciding the emptiness of certain semi-algebraic sets bounded by cubic polynomials, can be "lifted" to a purely combinatorial problem. We propose an effective algorithm for that problem, and use it to gain new insights into the structure of geometric permutations. [20]