On some problems of Euclidean Ramsey theory. (English) Zbl 1349.05334
Anal. Math. 41, No. 4, 299-310 (2015); erratum ibid. 42, No. 3, 295 (2016).
Summary: In the paper we prove, in particular, that for any measurable coloring of the euclidean plane with two colours there is a monochromatic triangle with some restrictions on the sides. Also we consider similar problems in finite fields settings.
MSC:
| 05D10 | Ramsey theory |
References:
| [1] | M. Bennett, D. Hart, A. Iosevich, J. Pakianathan, and M. Rudnev, Group actions and geometric combinatorics in Fdp, arXiv:1311.4788v1 [math.CO], 19 Nov 2013. |
| [2] | P. Cameron, J. Cilleruelo and O. Serra, On monochromatic solutions of equations in groups, Rev. Mat. Iberoam., 23(2007), 385-395. · Zbl 1124.05086 · doi:10.4171/RMI/499 |
| [3] | P. Erdos, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus, Euclidean Ramsey theorems, I, II, III, J. Combin. Theory Ser. A, 14(1973), 341-363; Coll. Math. Soc. Janos Bolyai, 10(1973) North Holland (Amsterdam, 1975), 529-557 and 559-583. · Zbl 0276.05001 · doi:10.1016/0097-3165(73)90011-3 |
| [4] | K. J. Falconer, The realization of distances in measurable subsets covering Rn, J. Combin. Theory A, 31(1981), 187-189. · Zbl 0469.05021 · doi:10.1016/0097-3165(81)90014-5 |
| [5] | D. Hart, A. Iosevich, Ubiquity of simplices in subsets of vector spaces over finite fields, Analysis Math., 34(2008), 29-38. · Zbl 1164.05064 · doi:10.1007/s10476-008-0103-z |
| [6] | A. Iosevich and D. Koh, Extension theorems for spheres in the finite field setting, Forum Math., 22(2010), no. 3, 457-483. · Zbl 1193.42068 · doi:10.1515/forum.2010.025 |
| [7] | L. Landau, Monotonicity and bounds on Bessel functions, mathematical physics and quantum field theory, Electron. J. Differential Equations, Conf. 04 (2000), 147-154. · Zbl 0976.33002 |
| [8] | A. F. Nikiforov, V. B. Uvarov, Special functions of mathematical physics, Birkhäuser (Basel, 1988). · Zbl 0624.33001 · doi:10.1007/978-1-4757-1595-8 |
| [9] | F. M. DE Oliveira Filho and F. Vallentin, Fourier analysis, linear programming, and densities of distance avoiding sets in Rn, arXiv:0808.1822v2 [math.CO] 2 Dec 2008. · Zbl 1205.90196 |
| [10] | A. Soifer, The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators, Springer Science & Business Media (New York, 2008). · Zbl 1221.05001 |
| [11] | A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A., 34(1948), 204207. · Zbl 0032.26102 · doi:10.1073/pnas.34.5.204 |
| [12] | J. Wolf, The minimum number of monochromatic 4-term progressions in Zp, J. Comb., 1(2010), 53-68. · Zbl 1227.11037 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
