<p>The following question was asked by Prendiville: given an <i>r</i>-colouring of the interval <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\{2, \dotsc , N\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>N</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, what is the minimum number of monochromatic solutions of the equation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$xy = z$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mi>z</mi> </mrow> </math></EquationSource> </InlineEquation>? For <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$r=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we show that there are always asymptotically at least <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(1/2\sqrt{2}) N^{1/2} \log N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <msqrt> <mn>2</mn> </msqrt> <mo stretchy="false">)</mo> </mrow> <msup> <mi>N</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo>log</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> monochromatic solutions, and that the leading constant is sharp. For <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$r=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$r=4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>=</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> we obtain tight results up to a multiplicative logarithmic factor. We also provide bounds for more colours and other multiplicative equations.</p>

On the number of monochromatic solutions to multiplicative equations

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2025年12月3日

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Lucas Aragão,
Jonathan Chapman,
Miquel Ortega,
Victor Souza

摘要

The following question was asked by Prendiville: given an r-colouring of the interval $\{2, \dotsc , N\}$ ${2, \dots, N}$ , what is the minimum number of monochromatic solutions of the equation $xy = z$ $x y = z$ ? For $r=2$ $r = 2$ , we show that there are always asymptotically at least $(1/2\sqrt{2}) N^{1/2} \log N$ $(1 / 2 \sqrt{2}) N^{1 / 2} log N$ monochromatic solutions, and that the leading constant is sharp. For $r=3$ $r = 3$ and $r=4$ $r = 4$ we obtain tight results up to a multiplicative logarithmic factor. We also provide bounds for more colours and other multiplicative equations.