On the minimum number of monochromatic solutions to the strict Schur inequality in 2-colored integer intervals with negative left endpoint
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910105687031808 |
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| author | Yang, Gang Liang, Jinxia Mao, Yaping Yang, Chenxu Zhang, Ayun |
| author_facet | Yang, Gang Liang, Jinxia Mao, Yaping Yang, Chenxu Zhang, Ayun |
| contents | Kosek, Robertson, Sabo, and Schaal studied the minimum number \(M_k(n)\) of monochromatic solutions to the strict Schur inequality system $x_1\le x_2\le x_3$ and $x_1+x_2<x_3$ in \(2\)-colorings of \([k+1,k+n]\). They proved that for every fixed \(k\ge 0\), $M_k(n)= \frac{n^3}{12(1+2\sqrt2)^2}(1+o_k(1)),$ and left open the case \(k\le -2\). In this paper, we resolve that remaining range. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04553 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the minimum number of monochromatic solutions to the strict Schur inequality in 2-colored integer intervals with negative left endpoint Yang, Gang Liang, Jinxia Mao, Yaping Yang, Chenxu Zhang, Ayun Combinatorics Kosek, Robertson, Sabo, and Schaal studied the minimum number \(M_k(n)\) of monochromatic solutions to the strict Schur inequality system $x_1\le x_2\le x_3$ and $x_1+x_2<x_3$ in \(2\)-colorings of \([k+1,k+n]\). They proved that for every fixed \(k\ge 0\), $M_k(n)= \frac{n^3}{12(1+2\sqrt2)^2}(1+o_k(1)),$ and left open the case \(k\le -2\). In this paper, we resolve that remaining range. |
| title | On the minimum number of monochromatic solutions to the strict Schur inequality in 2-colored integer intervals with negative left endpoint |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.04553 |