Tuesday, October 10, 2017
symmetric 2 variable function
Suppose S = \{1,2,\dots,n\} and f: S \times S \to R satisfies the following:
f(i,j) + f(j,i) = 0 \forall i,j \in S
Now for any two number i,j, we say that i is superior to j if there exists a k (not necessarily distinct from i,j) such that f(i,k) + f(k,j) \geq 0.
Show that there exists a number x \in S such that x is superior to all elements of S.
Labels:
Algebra,
Combinatorics,
function,
graph,
graph theory,
induction,
maximal principle,
strong induction
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