Activities This Week

A family of sets F is said to satisfy the (p,q)-property if among any p sets in F, some q have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any p > q > d there exists a constant c = c_d(p,q), such that any family of compact convex sets in R^d that satisfies the (p,q)-property, can be pierced by at most c points. Helly’s Theorem is equivalent to the fact that c_d(p,p)=1 (p > d).

In a celebrated result from 1992, Alon and Kleitman proved the conjecture. However, obtaining sharp bounds on the minimal such c_d(p,q), called `the Hadwiger-Debrunner numbers’, is still a major open problem in combinatorial geometry.

In this talk we present improved upper and lower bounds on the Hadwiger-Debrunner numbers, the latter using the hypergraph container method. Based on joint works with Shakhar Smorodinsky and Gabor Tardos.

קבוצה יוצרת $X$ של חבורה $G$ היא קבוצת איברים כך שכל איבר ב $G$ שווה למכפלה של איברים מ $X$ והופכיים של איברים מ $X$. בעיית היצירה עבור חבורה $G$ עוסקת בשאלה האם ניתן לקבוע, בהינתן תת קבוצה סופית $X$ של $G$, אם $X$ יוצרת את $G$. בעיית היצירה בחבורה $G$ פתירה, אם יש אלגוריתם שבהינתן תת קבוצה סופית $X$ של $G$ קובע אם $X$ יוצרת את $G$.

אנו נדון בבעיית היצירה במספר חבורות אינסופיות, ביניהן החבורה $F$ של תומפסון שניתנת להגדרה כחבורה של פונקציות על קטע היחידה $[0,1]$.

The monochromatic layers of the chromatic filtration on spectra, that is the K(n)-local (stable 00-)categories Sp_{K(n)} enjoy many remarkable properties. One example is the vanishing of the Tate construction due to Hovey-Greenlees-Sadofsky. The vanishing of the Tate construction can be considered as a natural equivalence between the colimits and limits in Sp_{K(n)} parametrized by finite groupoids. Hopkins and Lurie proved a generalization of this result where finite groupoids are replaced by arbitrary \pi-finite 00-groupoids.

There is another possible sequence of (stable 00-)categories who can be considered as “monochromatic layers”, those are the T(n)-local 00-categories Sp_{T(n)}. For the Sp_{T(n)} the vanishing of the Tate construction was proved by Kuhn. We shall prove that the analog of Hopkins and Lurie’s result in for Sp_{T(n)}. Our proof will also give an alternative proof for the K(n)-local case.

This is a joint work with Shachar Carmeli and Lior Yanovski