Rainbow trapezoids with given area
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917341668835328 |
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| author | Adhikari, Sukumar Das Naia, Tássio Serra, Oriol |
| author_facet | Adhikari, Sukumar Das Naia, Tássio Serra, Oriol |
| contents | A well-known result by Graham in Euclidean Ramsey Theory states that, for every positive real number $A$, every coloring of the plane with finite number of colors contains a monochromatic triangle of area $A$. We consider canonical versions of this result. We show that every $3$-coloring of the plane integer lattice contains either a rainbow triangle of area $1/2$ or a monochromatic rectangle of any given area whose sides are parallell to the axes. We also show that, under natural conditions, there are numbers $A$ and $B$ such that every coloring of the plane integer lattice contains either a monochromatic rectangle of area $A$ or a rainbow trapezoid of area $B$. As usual, only vertex colors are considered: e.g., a monochromatic rectangle is a set of four points in the lattice which a) are the vertices of a rectangle and b) are assigned the same color. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_13841 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rainbow trapezoids with given area Adhikari, Sukumar Das Naia, Tássio Serra, Oriol Combinatorics 05C55 (Primary), 05D10 (Secondary) A well-known result by Graham in Euclidean Ramsey Theory states that, for every positive real number $A$, every coloring of the plane with finite number of colors contains a monochromatic triangle of area $A$. We consider canonical versions of this result. We show that every $3$-coloring of the plane integer lattice contains either a rainbow triangle of area $1/2$ or a monochromatic rectangle of any given area whose sides are parallell to the axes. We also show that, under natural conditions, there are numbers $A$ and $B$ such that every coloring of the plane integer lattice contains either a monochromatic rectangle of area $A$ or a rainbow trapezoid of area $B$. As usual, only vertex colors are considered: e.g., a monochromatic rectangle is a set of four points in the lattice which a) are the vertices of a rectangle and b) are assigned the same color. |
| title | Rainbow trapezoids with given area |
| topic | Combinatorics 05C55 (Primary), 05D10 (Secondary) |
| url | https://arxiv.org/abs/2603.13841 |