Definition:

- Erdös is the only person with the Erdös Lap Number 0.

- In order to acquire the Erdös Lap Number i one has to sit on a lap of a person with the number i-1.

For example, Barbie Freidin (née Benzer) has the Erdös Lap Number of 1. On this photo (taken from the Fall 1996 issue of "Math PUrview") she is the leftmost epsilon. | |

Therefore, Oleg Pikhurko gets
the the Erdös Lap Number 2. On the photo he is the O(1) (the O big
of one) sitting on Barbie's lap. |

Open Question (Małgorzata Bednarska $10): Does there exist a person whose

Peter Griztmann conjectured that the answer is "Yes" and that János Pach is such a person. Unfortunately, even János himself was not able to produce a proof that his Erdös Lap Number is 1 (although he said that "it is a reasonable conjecture").

The following photo (posted by János Pach on Gil Kalai's blog) led some people to believe that they have a promising approach to the proof. However, Uncle Paul holds not János but his daughter...

Theorem (Ron Graham): The Erdös lap digraph contains a directed cycle!

Proof (presented by Ron at the Erdös Centennial Conference, 2013).

Erdos lap number 3 comes as an extra bonus for 21-484 "Graph Theory" students:

Sometimes I'm recognized even in public transport :-)