Erdős number
Mathematicians are assigned a number called the Erdős number (named after the famous mathematician Paul Erdős). Only Paul Erdős himself has an Erdős number of 0. Any mathematician who has written a research paper with Erdős has an Erdős number of 1. For other mathematicians: Suppose mathematician X has co-authored papers with several others. Among them, mathematician Y has the smallest Erdős number (y). Then, X's Erdős number is y + 1. If no co-authorship chain connects to Erdős, the Erdős number is ∞ (∞). Conference Scenario: A seven-day mini-conference was organized in memory of Paul Erdős, with eight mathematicians: A, B, C, D, E, F, G, H. Initial Conditions: At the start of the conference: A was the only participant with an infinite (∞) Erdős number. Nobody had an Erdős number less than F's (i.e., F had the smallest). Key Events: Day 3: F co-authored a paper with A and C. This reduced the average Erdős number of the group to 3. The Erdős numbers of B, D, E, G, H remained unchanged. No other co-authorship among any three members could have reduced the average to 3. At the end of Day 3: Five members had identical Erdős numbers. The other three had distinct Erdős numbers. Day 5: E co-authored a paper with F. This reduced the group’s average Erdős number by 0.5. The Erdős numbers of the other six remained unchanged. No other papers were written during the conference.
Who had an Erdös number of ∞ at the beginning of the conference?