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On monochromatic solutions to linear equations over the integers
Published 17 Oct 2024 in math.CO | (2410.13758v2)
Abstract: We study the number of monochromatic solutions to linear equations in a $2$-coloring of ${1,\ldots,n}$. We show that any nontrivial linear equation has a constant fraction of solutions that are monochromatic in any $2$-coloring of ${1,\ldots,n}$. We further study commonness of four-term equations and disprove a conjecture of Costello and Elvin by showing that, unlike over $\mathbb{F}_p$, the four-term equation $x_1 + 2x_2 - x_3 - 2x_4 = 0$ is uncommon over ${1,\ldots,n}$.
- Finding patterns avoiding many monochromatic constellations. Experiment. Math., 19(4):399–411, 2010.
- Avoiding monochromatic solutions to 3-term equations. J. Comb., 14(3):281–304, 2023.
- Boris A. Datskovsky. On the number of monochromatic Schur triples. Adv. in Appl. Math., 31(1):193–198, 2003.
- Common and Sidorenko linear equations. Q. J. Math., 72(4):1223–1234, 2021.
- On the distribution of monochromatic configurations. In Irregularities of partitions (Fertőd, 1986), volume 8 of Algorithms Combin. Study Res. Texts, pages 71–87. Springer, Berlin, 1989.
- On Schur properties of random subsets of integers. J. Number Theory, 61(2):388–408, 1996.
- From discrete to continuous: monochromatic 3-term arithmetic progressions. Math. Comp., 93(350):2959–2983, 2024.
- A removal lemma for systems of linear equations over finite fields. Israel Journal of Mathematics, 187:193–207, 2012.
- Monochromatic 4-term arithmetic progressions in 2-colorings of ℤnsubscriptℤ𝑛\mathbb{Z}_{n}blackboard_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. J. Combin. Theory Ser. A, 119(5):1048–1065, 2012.
- On the asymptotic minimum number of monochromatic 3-term arithmetic progressions. J. Combin. Theory Ser. A, 115(1):185–192, 2008.
- Richard Rado. Studien zur Kombinatorik. Math. Z., 36(1):424–470, 1933.
- A 2222-coloring of [1,n]1𝑛[1,n][ 1 , italic_n ] can have (1/22)n2+O(n)122superscript𝑛2𝑂𝑛(1/22)n^{2}+O(n)( 1 / 22 ) italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_O ( italic_n ) monochromatic Schur triples, but not less! Electron. J. Combin., 5:Research Paper 19, 4, 1998.
- Tomasz Schoen. The number of monochromatic Schur triples. European J. Combin., 20(8):855–866, 1999.
- On the minimum number of monochromatic generalized Schur triples. Electron. J. Combin., 24(2):Paper No. 2.20, 20, 2017.
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