Bar-Ilan Combinatorics Seminar

The next meeting of the seminar will take place, IYH,

(when)  Sunday, 24 Adar (March 18), 14:00-15:30

(where) Room 201 (Math & CS Seminar Room), Building 216, Bar-Ilan
University

(who)    Gili Golan (Bar-Ilan University)

(what)   Hindman's coloring theorem in arbitrary semigroups

Abstract:
Hindman's Theorem asserts that, for each finite coloring of $\N$,
there are distinct $a_1, a_2, \dots \in \N$ such that all sums
$a_{i_1} + a_{i_2} + \dots + a_{i_m}$
($m \ge 1$, $i_1 < i_2 < \dots < i_m$) have the same color.
Shevrin's classification of semigroups and a proof of Hindman's
Theorem, due to Galvin and Glazer, imply together that, for each
infinite semigroup $S$, there are distinct $a_1, a_2, \dots \in S$
such that all but finitely many of the products
$a_{i_1}a_{i_2}\cdots a_{i_m}$ ($m \ge 1$, $i_1 < i_2 < \dots < i_m$)
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.

Simple proofs. No background is needed.

Joint work with Boaz Tsaban.

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You are all invited! Graduate students are especially welcome.