用超滤子表示Szemerédi定理
Formulating Szemerédi's Theorem in Terms of Ultrafilters
摘要：
范德瓦尔登定理断言，如果你用五种不同的颜色给自然数着色，那么你总是可以找到任意长的具有相同颜色的数字序列，它们形成一个等差数列。Szemerédi定理推广了这一说法，并断言每一个具有正密度的自然数子集都包含任意长度的等差数列。范德瓦尔登定理可以用初等组合学证明，但也可以用超滤子和斯通-切赫紧化βN给出解释和简短证明。这篇毕业论文给出了一个解释Szemerédi定理方面的超滤波器以及。特别地，van der Waerden定理等价于存在一个具有特殊性质的超滤子，我们将证明Szemerédi定理等价于这样一个事实，即关于βN上的计数测度，几乎所有的超滤子都具有这些性质。
英文摘要：
Van der Waerden's theorem asserts that if you color the natural numbers with, say, five different colors, then you can always find arbitrarily long sequences of numbers that have the same color and that form an arithmetic progression. Szemerédi's theorem generalizes this statement and asserts that every subset of natural numbers with positive density contains arithmetic progressions of arbitrary length. Van der Waerden's theorem can be proven using elementary combinatorics, but it is also possible to give an interpretation and a short proof in terms of ultrafilters and the Stone-Čech compactification βN. This diploma thesis gives an interpretation of Szemerédi's theorem in terms of ultrafilters as well. In particular, van der Waerden's theorem is equivalent to the existence of a single ultrafilter with special properties and we will show that Szemerédi's theorem is equivalent to the fact that with respect to a counting measure on βN, almost all ultrafilters have these properties.
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