Abstract
This chapter contains descriptions of 11 great theorems published in the Annals of Mathematics in 2005.
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- 1.
Indeed, \(f_a(x)<x\) is equivalent to \(a - x^2 < x\), or \(x^2+x>a\), or \(x^2+x+1/4 > a+1/4\), or \((x+1/2)^2 > a + 1/4\), which is obvious for any \(a<-1/4\), because the left-hand side is non-negative, while the right-hand side is negative.
- 2.
For any sequence \(a_N\), \(\limsup \limits _{N\rightarrow \infty } a_N := \lim \limits _{N\rightarrow \infty } (\!\sup \limits _{m \ge N} a_m)\).
- 3.
If you would like to see a proof, here it is. If A contains 5, it can contain 4 or 6 but not both, and similarly 3 or 7, 2 or 8, and 1 or 9, so no more than 5 elements in total. So we can assume that \(5\not \in A\). Because A can have at most 2 elements out of \(\{1,2,3\}\), and at most 2 out of \(\{7,8,9\}\), it may have 6 elements only if it contains 4 and 6. But then it excludes 2 and 8, hence it must be \(\{1, 3, 4, 6, 7, 9\}\), which is also impossible due to the progression (1, 4, 7), a contradiction.
- 4.
The definition of the \(n\times n\) determinant is also a bit complicated. Let \(S_n\) be the set of all possible permutations \(\sigma =(\sigma (1), \dots , \sigma (n))\) of the set \((1,2,\dots , n)\). For example, (3, 1, 2) is a possible permutation of (1, 2, 3). For general n, there are n! possible permutations. Each permutation \(\sigma \) can be “implemented” by starting from \((1,2,\dots , n)\) and exchange adjacent elements, e.g. \((1,2,3)\rightarrow (1,3,2)\rightarrow (3,1,2)\). The sign of sigma is defined as \((-1)^n\), where n is the number of steps in this sequence. The determinant of a matrix with entries \(c_{ij}\) is defined as \(\sum _{\sigma \in S_n} \text {sign}(\sigma )\prod _{i=1}^n c_{\sigma (i)i}\).
- 5.
For example, for 5 people such that exactly (1, 2), (2, 3), (3, 4), (4, 5), and (5, 1) are enemies, no 3 form a triangle, but 2 colours is not enough.
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Grechuk, B. (2019). Theorems of 2005. In: Theorems of the 21st Century. Springer, Cham. https://doi.org/10.1007/978-3-030-19096-5_5
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DOI: https://doi.org/10.1007/978-3-030-19096-5_5
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