Speaker: Shakhar Smorodinsky (BGU) Title: Improved bounds on the Hadwiger-Debrunner numbers Abstract: --------- The classical Helly's theorem says that if in a family of compact convex sets in $\Re^d$ every $d+1$ members have a non-empty intersection then the whole family has a non-empty intersection. In an attempt to generalize Helly's theorem, in 1957 Hadwiger and Debrunner posed an important conjecture that was proved more than 30 years later in a celebrated result of Alon and Kleitman: For a pair of integers p,q (p \geq q > d) there exists a constant C=C(p,q,d) such that the following holds: if in a family of convex sets out of every p members some q intersect then the whole family can be pierced with C points. It remains a wide open problem to provide sharp bounds on C(p,q,d). In this talk we show how to improve (already for $d=2$ and $q=3$) the previously best known bound (of $\tilde{O}(p^{d^2+d})$) provided in Alon and Kleitman's proof (for the case $q=d+1$). Time permitting, we also discuss several related problems. Joint work with Gabor Tardos