Some new results in multiplicative and additive Ramsey theory

Abstract

There are several notions of largeness that make sense in any semigroup, and others such as the various kinds of density that make sense in sufficiently well-behaved semigroups including <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-struck upper N comma plus right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {N},+)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-struck upper N comma dot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {N},\cdot )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It was recently shown that sets in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are <italic>multiplicatively large</italic> must contain arbitrarily large <italic>geoarithmetic progressions</italic>, that is, sets of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace r Superscript j Baseline left-parenthesis a plus i d right-parenthesis colon i comma j element-of StartSet 0 comma 1 comma ellipsis comma k EndSet right-brace"> <mml:semantics> <mml:mrow> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">{</mml:mo> </mml:mrow> </mml:mstyle> <mml:msup> <mml:mi>r</mml:mi> <mml:mi>j</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mspace width="negativethinmathspace"/> <mml:mo>+</mml:mo> <mml:mspace width="negativethinmathspace"/> <mml:mi>i</mml:mi> <mml:mi>d</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="negativethinmathspace"/> <mml:mo>:</mml:mo> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>∈</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">}</mml:mo> </mml:mrow> </mml:mstyle> </mml:mrow> <mml:annotation encoding="application/x-tex">\big \{r^j(a\!+\!id)\!:i,j\in \{0,1,\dotsc ,k\}\big \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, as well as sets of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace b left-parenthesis a plus i d right-parenthesis Superscript j Baseline colon i comma j element-of StartSet 0 comma 1 comma ellipsis comma k EndSet right-brace"> <mml:semantics> <mml:mrow> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">{</mml:mo> </mml:mrow> </mml:mstyle> <mml:mi>b</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mi>d</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>j</mml:mi> </mml:msup> <mml:mo>:</mml:mo> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>∈</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">}</mml:mo> </mml:mrow> </mml:mstyle> </mml:mrow> <mml:annotation encoding="application/x-tex">\big \{b(a+id)^j:i,j\in \{0,1,\dotsc ,k\}\big \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Consequently, given a finite partition of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, one cell must contain such configurations. In the partition case we show that we can get substantially stronger conclusions. We establish some combined additive and multiplicative Ramsey theoretic consequences of known algebraic results in the semigroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis beta double-struck upper N comma plus right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>β</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\beta \mathbb {N},+)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis beta double-struck upper N comma dot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>β</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\beta \mathbb {N},\cdot )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, derive some new algebraic results, and derive consequences of them involving geoarithmetic progressions. For example, we show that given any finite partition of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there must be, for each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, sets of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace b left-parenthesis a plus i d right-parenthesis Superscript j Baseline colon i comma j element-of StartSet 0 comma 1 comma ellipsis comma k EndSet right-brace"> <mml:semantics> <mml:mrow> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">{</mml:mo> </mml:mrow> </mml:mstyle> <mml:mi>b</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mi>d</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>j</mml:mi> </mml:msup> <mml:mo>:</mml:mo> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>∈</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">}</mml:mo> </mml:mrow> </mml:mstyle> </mml:mrow> <mml:annotation encoding="application/x-tex">\big \{b(a+id)^j:i,j\in \{0,1,\dotsc ,k\}\big \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> together with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the arithmetic progression <inline-formula content-type="math/mathml"> <mml:math