CAT 2006 — DILR Question 6

Answer the following question based on the information given below.

Mathematicians are assigned a number called Erdös number, (named after the famous mathematician, Paul Erdös). Only Paul Erdös himself has an Erdös number of zero. Any mathematician who has written a research paper with Erdös has an Erdös number of 1. For other mathematicians, the calculation of his/her Erdös number is illustrated below:

Suppose  that a mathematician X has co-authored papers with several other mathematicians. From  among them, mathematician Y has the smallest Erdös number. Let the Erdös number of Y be y. Then X has an Erdös number of y + 1. Hence any mathematician with no co-authorship chain connected to Erdös has an Erdös number of infinity.

• In a seven day long mini-conference organized in memory of Paul  Erdös, a close group of eight mathematicians, call them A, B, C, D, E, F, G and H, discussed some  research problems. At the beginning of the conference, A was the only participant who had an infinite Erdös number. Nobody had an Erdös number less than that of F.  
• On the third day of the conference F co-authored a paper jointly with A and C. This reduced the average Erdös  number  of  the group  of eight  mathematicians to 3. The Erdös numbers of B, D, E, G and H remained  unchanged with the writing of this paper. Further, no other co-authorship among any three members would have reduced the average Erdös number of the group of eight to as low as 3.  
• At the end of the third day, five members of this group had identical Erdös numbers while the other three had Erdös numbers distinct from each other.
• On the fifth day, E co-authored a paper with F which reduced the group‘s average Erdös number by  0.5. The Erdös numbers of the remaining six were unchanged with the writing of this paper. 
• No other paper was written during the conference.