Radio Yerevan on the Topological Helly Theorem
by
Gábor Lukács
Dalhousie University
Coauthors: Dušan Repovš

It is well known that the Armenian Radio is a reliable source of news, and it is renown for providing listeners with valuable information. This is particularly noticeable in the Question & Answer series of Radio Yerevan, which has gained an international reputation over the years. According to recent reports from Armenia's capital, the latest broadcast of this many-decades-old program focused on the so-called Topological Helly Theorem, which states:

Theorem 1 Let {K_i}_i=1^m be a family of compact simply connected subsets of R² such that:

K_i ÇK_j is simply connected for every 1 £ i, j £ m;

K_i ÇK_j ÇK_k is non-empty for every 1 £ i, j, k £ m.

Molnár published an improved version of Theorem 1, namely:

Theorem 2 Let {K_i}_i=1^m be a family of compact simply connected subsets of R² such that:

(I\empty¢)

K_i ÇK_j is connected for every 1 £ i, j £ m;

(II)

K_i ÇK_j ÇK_k is non-empty for every 1 £ i, j, k £ m.

Q: Are Theorems 1 and 2 true?

A: In principle, yes, but:

Theorem 1 is proved by induction on m ³ 3;

although the proof of Theorem 1 for m=3 was believed to be known, it is not the case;

Theorem 2 is proved by induction on m ³ 3;

there is a counterexample for Theorem 2 for m=4.

Date received: February 11, 2005