If 9 people are seated in a row of 12 chairs, then at least one consecutive set of 3 chairs is occupied by people.
(In reply to re: Sparseness .....spoiler
by Ady TZIDON)
I was thinking along the lines better expressed by Justin. The "sparseness" I had in mind was that attempting to space the nine Ss around the three Es would find the worst case (for the lemma) in SSESSESSESS which given three spaced Es would allow only 8 Ss before a group of three or more would be needed. (Justin, I believe, was making the same point in saying the average of the S subgroups must be greater than 2 (i.e. at least 3 in one group if 12 total seats, since presumably no fractional people allowed, nicht wahr?). I do not understand your "Why?" question to me. If any of the Es were adjacent, the number of allowable Ss without three or more in a row would be even less than 8; if the three Es were together, the maximum number of Ss would be only four SSEEESS. As the Razor (almost) goes, do not multiply "proofs" beyond necessity.