Hindmanʼs coloring theorem in arbitrary semigroups
Hindmanʼs coloring theorem in arbitrary semigroups
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任意半群中的 Hindman 着色定理
DOI：
10.1016/j.jalgebra.2013.08.007
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发表时间：
2013
影响因子：
0.9
通讯作者：
B. Tsaban
中科院分区：
文献类型：
--
作者：
Gili Golan;B. Tsaban
Hindman定理认为，对于自然数的每一个有限着色，都有不同的自然数a1，a2，...使得所有的和ai 1+ ai 2+ ai + ai m（m ∈ 1，i 1< i 2< i m）都有相同的颜色。著名的Galvin-Glazer对Hindman定理的证明和Shevrin对半群的分类一起暗示，对于每个无限半群S的每个有限着色，存在S的不同元素a1，a2，.，使得所有的乘积ai 1 ai 2 <$ai m（m <$1，i 1< i 2< i m）具有相同的颜色。利用这些方法，我们刻画了半群S，使得对于S的每一个有限着色，存在S的无限子半群T，使得T的除1/2个成员外的所有成员具有相同的颜色。我们的刻画将我们的研究与Milliken、伯恩赛德群和Tarski Monsters的经典问题联系起来。我们还提出了拉姆齐图着色定理在谢夫林理论中的应用。
Hindmanʼs Theorem asserts that, for each finite coloring of the natural numbers, there are distinct natural numbers a 1, a 2,… such that all of the sums a i 1+ a i 2+⋯+ a i m (m⩾ 1, i 1< i 2<⋯< i m) have the same color. The celebrated Galvin–Glazer proof of Hindmanʼs Theorem and a classification of semigroups due to Shevrin, imply together that, for each finite coloring of each infinite semigroup S, there are distinct elements a 1, a 2,… of S such that all but finitely many of the products a i 1 a i 2⋯ a i m (m⩾ 1, i 1< i 2<⋯< i m) have the same color. Using these methods, we characterize the semigroups S such that, for each finite coloring of S, there is an infinite subsemigroup T of S, such that all but finitely many members of T have the same color. Our characterization connects our study to a classical problem of Milliken, Burnside groups and Tarski Monsters. We also present an application of Ramseyʼs graph-coloring theorem to Shevrinʼs theory.