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You are essentially asking about the graph isomorphism problem. Or, more precisely, you are asking about canonical representation of graphs. There is a huge literature about this. See https://en.wikip …

Just playing around with it: The RHS multiplied by $n$ is the same as $$2 \sum_{k=0}^{n-1} \binom{n+1}{k} \binom{n}{k} \binom{n+1}{k+2}.$$ Subtracting this from $n$ times the LHS gives $$\sum_{k=0}^{n …

See Tao's book on Hilbert's fifth problem -- https://terrytao.wordpress.com/books/hilberts-fifth-problem-and-related-topics/ -- Theorem 3.1.7, where it is referred to as local Cartan's theorem.

Is it true that if $G$ has full spectrum with respect to a series $\sum_{n=1}^\infty a_n$ then $G$ has full spectrum also with respect to any other conditionally convergent series $\sum_{n=1}^\inft …

I think the answer to your question "do we have some good estimates for $f(n)$?" is no. Or at least, the situation is the same as that for triangle removal itself. The assertion that $f(n) = o(n^2)$ i …

The keywords I was looking for were "fictitious play" and "Brown--Robinson process". The short answer seems to be that it works, sometimes. For more information see https://en.wikipedia.org/wiki/Ficti …

Suppose you and I are playing a two-player zero-sum game repeatedly. There is a payoff matrix $(A_{ij})$, and if I play action $i$ and you play action $j$ then I receive $A_{ij}$ and you receive $-A_{ …

I have no reference for this problem, but let's at least write down the trivial bounds. Let $s_1,\dots s_r\in S^n$ and suppose $n>|S|^r$. Associate to each index $i$ the element $$(\pi_i(s_1),\dots,\ …

We can reduce Lucia's upper bound of $3/8$ a little further as follows. Begin by taking the $n/4$ geometric progressions of common ratio $2$ beginning at each odd number at most $n/2$. Then for each o …

Grunbaum, convex polytopes: http://books.google.co.uk/books?id=ISHO86XJ1CsC&printsec=frontcover#v=onepage&q&f=false