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{\Large המחלקה למתמטיקה, בן-גוריון}

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\textbf{ב}\emph{יום שלישי, 14 ביוני, 2016}
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\textbf{בשעה} \emph{14:30 -- 15:30}
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\textbf{ב}\emph{Math -101}

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ההרצאה

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{\Large\bfseries Improved bounds on the Hadwiger Debrunner numbers\par}
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תינתן על-ידי
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{\large\scshape  Shakhar Smorodinsky 
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  (BGU)
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\textbf{תקציר:}
  The classical Helly`s theorem states that if in a family of compact convex sets in R\^{}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 a conjecture that was proved more than 30 years later in a celebrated result of Alon and Kleitman: For any p,q (p \textgreater{}= q \textgreater{} d) there exists a constant C=C(p,q,d) such that the following holds: If in a family of compact convex sets, out of every p members some q intersect, then the whole family can be pierced with C points. Hadwiger and Debrunner themselves showed that if q is very close to p, then \$C=p-q+1\$ suffices.

The proof of Alon and Kleitman yields a huge bound \$C=O(p\^{}\{d\^{}2+d\})\$, and providing sharp upper bounds on the minimal possible C remains a wide open problem.

In this talk we show an improvement of the best known bound on C for all pairs \$(p,q)\$. In particular, for a wide range of values of q, we reduce C all the way to the almost optimal bound p-q+1\textless{}=C\textless{}=p-q+2. This is the first near tight estimate of C since the 1957 Hadwiger-Debrunner theorem.

Joint work with Chaya Keller and Gabor Tardos.
  


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