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Off-hand, I'm not so sure about the above comment. Sperner proved two lemmas on his way to Lebesgue's theorem. The second one is in Lyusternik Convex Figures and Polyhedra, along with Sperner's proof of Lebesgue's theorem. I'd have to find the reference.
In fact, it's known as Sperner's theorem, not lemma, see Sperner family and my comments in Talk:antichain. Perhaps a disambiguation remark at the top would be appropriate.--192.35.35.36 00:15, 18 Feb 2005 (UTC)
Please note that the dimension of the whole simplex is n, not k. k is a free variable; it can be anything less than n. So offsetting its value by one makes no difference. —David Eppstein (talk) 16:25, 3 October 2012 (UTC)[reply]
Yes, I understand. Maybe I'm missing something, but for any fixed k, 1<=k<=n, the k points here form a k-1 dimensional simplex/subface, I think. Joerg Bader (talk) 16:43, 3 October 2012 (UTC)[reply]
Ah, it's a mismatch between the number of points and the dimension of the subspace. Another way to fix it would be to start indexing the points at 0, I suppose... or, we don't really need the extra variable k because the condition only needs to be true for the (n-1)-dimensional faces (and the rest follow automatically). —David Eppstein (talk) 16:58, 3 October 2012 (UTC)[reply]
What is the "degree of a triangle"? I was able to find a definition of the degree of a vertex, but no luck with the degree of a triangle. Without understanding the term, I do not find it to be "easily seen . . ." Thanks. Maurice Fox (talk) 00:56, 11 May 2014 (UTC)[reply]
Can someone help clear the ambiguity in this for me? what is capital A and B the proof is referring to? is it the regions "small" a and b ? also doesn't this proof need some citations? — Preceding unsigned comment added by MVDVPRSD (talk • contribs) 14:35, 29 June 2023 (UTC)[reply]
Ok.. it seem to be referring to the vertices of the triangle; if someone could label the image and find a citation for this proof that'd be grt MVDVPRSD (talk) 12:07, 30 June 2023 (UTC)[reply]