Tuesday, Nov. 14 |

09:00-10:00 |
**Andrew Suk: **Andrew Suk (UC San Diego)
Short edges in topological graphs [+]
Let $h(n)$ be the minimum integer such that every complete $n$-vertex simple topological graph contains an edge that crosses at most $h(n)$ other edges. In 2009, Kyčll and Valtr showed that $h(n)=O(n^2/log^{1/4}n)$, and in the other direction, gave constructions showing that $h(n)=\Omega(n^{3/2})$. In this talk, I will sketch a proof showing that $h(n)=O(n^{7/4})$. The proof is based on a variant of Chazelle-Welzl's matching theorem for set systems with bounded VC-dimension, which may be of independent interest. I will also discuss several other related problems. |

Coffee break |

10:30-11:30 |
**Karim Adiprasito: **Karim Adiprasito (Hebrew University, Jerusalem)
Maps, manifolds and metrics: a panorama of combinatorial construction [+]
I will survey and introduce a handy collection of combinatorial constructions and use them to construct interesting examples and counterexamples to conjectures all around geometry. |

11:30-12:30 |
**Shakhar Smorodinsky: **Shakhar Smorodinsky (Ben Gurion Univ., Be’er Sheva)
An extension of $\varepsilon$-nets for hypergraphs with bounded VC-dimension [+]
We study the following generalization of the notion of $\varepsilon$-nets in hypergraphs. Let $H=(V,E)$ be a hypergraph, $0 < \varepsilon \leq 1$ a real, and $t$ a positive integer. We call a family $N$ of $t$-element subsets of $V$ an $\varepsilon$-$t$-net if every hyperedge $S$ in $E$ with size at least $\varepsilon |V|$ contains at least one set from $N$. When $t=1$ this is the classical notion of $\varepsilon$-net. We show that for any constant integers $t,d$ any hypergraph with VC-dimension $d$ admits an $\varepsilon$-$t$-net of size $O((d/\varepsilon) \log(1/\varepsilon))$. Joint work with Noga Alon, Bruno Jartoux, Chaya Keller and Yelena Yuditsky. |

Lunch break |

14:00-15:00 |
**Zoltán Füredi: **Zoltán Füredi (Rényi Institute, Budapest)
Extremal hypergraphs in combinatorial geometry [+]
Ever since Erdős and Szekeres used Ramsey's theorem to find a convex $n$-gon in all sufficiently large planar point sets there have been many applications of extremal graph and hypergraph theory. Here we overview some classical and newer results, in particular when Turán theory is used. |

Coffee break |

15:30-16:30 |
**Rom Pinchasi: **Rom Pinchasi (Technion, Haifa)
On digons in planar arrangements of pairwise intersecting circles and pseudo-circles [+]
Given an arrangement of pairwise intersecting pseudo-circles, digons are the faces in this arrangement that have two edges. We will address an old conjecture of Grünbaum from 1972 about the maximum number of digons in arrangements of $n$ pairwise intersecting pseudo-circles in the plane. In particular, we settle Grünbaum's conjecture in the special case of geometric circles and show that the number of digons there is at most $2n-2$. We also present tight upper bounds on the number of lunes and lenses, separately, in arrangements of pairwise intersecting circles and pseudo-circles. The solution of Grünbaum's conjecture for circles is a joint work with Eyal Ackerman, Gábor Damásdi, Eric Gottlieb, Balázs Keszegh, and Rebeka Raffay. |

16:30-17:30 |
**József Solymosi: **József Solymosi (UBC, Vancouver)
On the structure of extremal point-line arrangements [+]
A classical result of Szemerédi and Trotter gives an upper bound on the number of point-line incidences. In particular, between $n$ points and $n$ lines, the number of incidences is $O(n^{4/3})$. We show that if an arrangement has many incidences, it determines $Omega(n^{4/3})$ incidences, then one can select $O(1)$, a constant number of points such that by fixing them, a large fraction of the arrangement becomes rigid. This is a continuation of our work with Zeev Dvir, Ankit Garg, and Rafael Oliveira, analyzing point sets with many collinear triples. |

17:30-18:30 |
**Chaya Keller: **Chaya Keller (Ariel University)
New Bounds for Zarankiewicz's Problem via $\varepsilon$-$t$-nets [+]
The classical Zarankiewicz's problem asks for the maximum number of edges in a bipartite graph on $n$ vertices which does not contain the complete bipartite graph $K_{t,t}$. In one of the cornerstones of extremal graph theory, Kővári, Sós and Turán proved an upper bound of $O(n^{2-\frac{1}{t}})$. In a celebrated result, Fox, Pach, Sheffer, Suk and Zahl obtained an improved bound of $O(n^{2-\frac{1}{d}})$ for graphs of VC-dimension $d$ (where $d < t$). Recently, Chan and Har-Peled presented (quasi-)linear upper bounds for several classes of geometrically-defined incidence graphs, including a bound of $O(n \log \log n)$ for the incidence graph of points and pseudo-discs in the plane.
In this talk we present a new approach to Zarankiewicz's problem, via $\varepsilon$-$t$-nets — a recently introduced generalization of the classical notion of $\varepsilon$-nets. We show that the existence of `small'-sized $\varepsilon$-$t$-nets implies upper bounds for Zarankiewicz's problem. Using the new approach, we obtain a new proof of the $O(n^{2-\frac{1}{d}})$ bound of Fox et al., and a sharp bound of $O(n)$ for the intersection graph of two families of pseudo-discs, thus both improving and generalizing the result of Chan and Har-Peled from incidence graphs to intersection graphs. We also obtain improved bounds for several other classes of geometric intersection graphs, including a sharp $O(n\frac{\log n}{\log \log n})$ bound for the intersection graph of two families of axis-parallel rectangles. Joint work with Shakhar Smorodinsky. |