Nonstandard methods for Ramsey theory

Current topics of research

  • Study of properties of monads of arbitrary sets and of tensor products of arbitrary ultrafilters, in progress.
  • Combinatorial properties of linear combinations of idempotent ultrafilters, in progress (joint with Mauro Di Nasso).
  • Refinements of the notions of finite embeddabilities, in progress (joint with Yunied Puig de Dios).

List of relevant literature

  • B. Barber, N. Hindman, I. Leader, and D. Strauss, Partition regularity without the columns property, Proc. Amer. Math. Soc. 143 (2015), 3387--3399.
  • M. Beiglböck, V. Bergelson, N. Hindman, D. Strauss, Some new results in multiplicative and additive Ramsey theory, T. Am. Math. Soc., Vol. 360 (2008), No. 2, 819–847.
  • M. Beiglböck, An ultrafilter approach to Jin’s Theorem, Israel J. Math., Vol. 185 (2011), Issue 1, 369–374.
  • V. Benci, M. Di Nasso, Alpha-theory: an elementary axiomatic for nonstandard analysis, Expo. Math., Vol. 21 (2003), 355–386.
  • V. Benci, L. Luperi Baglini, A non-archimedean algebra and the Schwartz impossibility theorem, Monatsh. Math., Vol. 176 (2015), 503–520, doi: 10.1007/s00605-014-0647-x.
  • V. Bergelson, Ergodic Ramsey Theory - un update, in "Ergodic Theory of $\Z^d$-actions" (M. Pollicott and K. Scmidt, eds.), London Math. Soc. Lecture Notes Series 228 (1996), 1--61.
  • V. Bergelson, Ultrafilters, IP sets, dynamics, and combinatorial number theory, in "Ultrafilters Across Mathematics", (V. Bergelson, A. Blass, M. Di Nasso and R. Jin, eds.), Contemp. Math. 530, AMS (2010), 23--47.
  • V. Bergelson, H. Furstenberg and R. McCutcheon, IP-sets and polynomial recurrence, Ergodic Theory Dynam. Systems 16 (1996), no. 5, 963--974.
  • V. Bergelson, J.H. Johnson jr., and J. Moreira, New polynomial and multidimensional extensions of classical partition results, J. Combin. Theory Ser. A., 147 (2017), 119--154.
  • V. Bergelson and J. Moreira, Ergodic theorem involving additive and multiplicative groups of a field and $\{x+y,xy\}$ patterns, Ergodic Theory Dynam. Systems, 37 (2017), 673--692.
  • V. Bergelson and J. Moreira, Measure preserving actions of affine semigroups and $\{x+y, xy\}$ patterns, Ergodic Theory Dynam. Systems, 1--26, doi:10.1017/etds.2016.39.
  • A. Blass, M. Di Nasso, Finite embeddability of sets and ultrafilters, Bull. Pol. Acad. Sci. Math. 63 (2015), 195–206.
  • T. Brown, Monochromatic solutions of exponential equations, Integers, Vol. 15A (2015), A2.
  • T.C. Brown and V. Röedl, Monochromatic solutions to equations with unit fractions, Bull. Aust. Math. Soc. 43 (1991), 387--392.
  • P. Csikvari, K. Gyarmati and A. Sarközy, Density and Ramsey type results on algebraic equations with restricted solution sets, Combinatorica 32, Issue 4 (2012), 425--449.
  • D. Davenport, N. Hindman, I. Leader, and D. Strauss, Multiply partition regular matrices, Discrete Math. 322 (2014), 61--68.
  • W. Deuber, Partitionen und lineare Gleichungssysteme, Math. Z. 133 (1973), 109--123.
  • W. Deuber, N. Hindman, I. Leader and H. Lefmann, Infinite partition regular matrices, Combinatorica 15, Issue 3 (1995), 333--355.
  • M. Di Nasso, Embeddability properties of difference sets, Integers, Vol. 14 (2014), A-27.
  • M. Di Nasso, Intersections of sets of distances, Integers, Vol. 16 (2016), A85.
  • M. Di Nasso, Iterated hyper-extensions and an idempotent ultrafilter proof of Rado's Theorem, Proc. Amer. Math. Soc. 143 (2015), 1749--1761.
  • M. Di Nasso, Hypernatural numbers as ultrafilters, Chapter 11 of "Nonstandard Analysis for the Working Mathematician", (P.A. Loeb and M. Wolff, eds.), 2nd edition, Springer, 2015.
  • M. Di Nasso, A taste of nonstandard methods in combinatorics of numbers, in "Geometry, Structure and Randomness in Combinatorics'' (J. Matousek, J. Nešetřil, M. Pellegrini, eds.), CRM Series, Scuola Normale Superiore, Pisa, 2015.
  • M. Di Nasso and M. Lupini, Nonstandard analysis and the sumset phenomenon in arbitrary amenable groups, Illinois J. Math. 58 (2014), 11--25.
  • M. Di Nasso, I. Goldbring, R. Jin, S. Leth, M. Lupini, and K. Mahlburg, High density piecewise syndeticity of sumsets, Adv. Math. 278 (2015), 1--33.
  • M. Di Nasso, I. Goldbring, R. Jin, S. Leth, M. Lupini, and K. Mahlburg, On a sumset conjecture of Erdos, Canad. J. Math. 67 (2015), 795-809.
  • M. Di Nasso, I. Goldbring, R. Jin, S. Leth, M. Lupini, and K. Mahlburg, Approximate polynomial structure in additively large sets, Integers 16 (2016), Article A49.
  • M. Di Nasso, I. Goldbring, R. Jin, S. Leth, M. Lupini, and K. Mahlburg, A monad measure space for logarithmic density, Monatsh. Math. 181 (2016), 577--599.
  • M. Di Nasso, I. Goldbring, R. Jin, S. Leth, M. Lupini, and K. Mahlburg, Approximate polynomial structure in additively large sets, Integers 16 (2016), A49.
  • M. Di Nasso and M. Riggio, Fermat-like equations that are not partition regular, Combinatorica (2017). <a href="https://doi.org/10.1007/s00493-016-3640-2arXiv:1605.07527" target="_blank">doi.org/10.1007/s00493-016-3640-2arXiv:1605.07527</a> (2016).
  • M. Di Nasso, L. Luperi Baglini, Ramsey properties of nonlinear Diophantine equations, Adv. Math., in press, arXiv:1606.02056.
  • N. Frantzikinakis and B. Host, Higher order Fourier analysis of multiplicative functions and applications, J. Amer. Math. Soc., to appear. (Published electronically: March 1, 2016.)
  • R. Goldblatt, Lectures on the Hyperreals -- An Introduction to Nonstandard Analysis, Graduate Texts in Mathematics 188, Springer, New York, 1998.
  • R. Graham, B. Rothschild and J. Spencer, Ramsey Theory (2nd edition), Wiley, New York, 1990.
  • R. Graham, Some of my favorite problems in Ramsey theory, Comb. Number Theory 7(2) (2007), 229--236.
  • B. Green and T. Sanders, Monochromatic sums and products, Discrete Analysis (2016).
  • M.J.H. Heule, O. Kullmann and V.W. Marek, Solving and verifying the boolean Pythagorean Triples problem via Cube-and-Conquer, In: Creignou N., Le Berre D. (eds), Theory and Applications of Satisfiability Testing – SAT 2016, Lecture Notes in Computer Science, 9710, Springer, Cham.
  • N. Hindman, Partitions and sums and products of integers, Trans. Amer. Math. Soc. 247 (1979), 227--245.
  • N. Hindman and I. Leader, Nonconstant Monochromatic Solutions to Systems of Linear Equations, in Topics in Discrete Mathematics, Springer, Berlin (2006), 145--154.
  • N. Hindman, Monochromatic Sums Equal to Products in $\mathbb{N}$, Integers 11A (2011), Article 10, 1--10.
  • N. Hindman and D. Strauss, Algebra in the Stone-Cech Compactification: Theory and Applications, 2nd edition, de Gruyter, Berlin, 2011.
  • J. Hirschfeld, Nonstandard combinatorics, Studia Logica, Vol. 47 (1988), No. 3, 221–232.
  • R. Jin, Proc. Amer. Math. Soc. 130 (2002), 855--861.
  • R. Jin, Applications of nonstandard analysis in additive number theory, B. Symb. Log., Vol. 6 (2000), No. 3, 331–341.
  • A. Khalfalah and E. Szemerédi, On the number of monochromatic solutions of $x+y=z^2$, Comb. Prob. and Comp. 15 (2006), 213--227.
  • H. Lefmann, On partition regular systems of equations, J. Combin. Theory Ser. A, 58 (1991), 35--53.
  • S.C. Leth, Some Nonstandard Methods in Combinatorial Number Theory, Studia Logica, Vol. 47 (1988), No. 3, 265–278.
  • S.C. Leth, Nonstandard methods and the Erdős-Turán conjecture, in: The Strength of Nonstandard Analysis (I. Van den Berg, V. Neves, eds.), Springer (2006), 133–142.
  • L. Luperi Baglini, A nonstandard technique in combinatorial number theory, European J. Combin., Vol. 48 (2015), 71–80, doi: 10.1016/j.ejc.2015.02.010.
  • L. Luperi Baglini, Partition regularity of nonlinear polynomials: a nonstandard approach, Integers, Vol. 14 (2014), A-30.
  • L. Luperi Baglini, Ultrafilters maximal for finite embeddability, J. Log. Anal., Vol. 6 (2014), A-6, 1–16, doi: <a href="http://dx.doi.org/10.4115/jla.v6i0.226" target="_blank">dx.doi.org/10.4115/jla.v6i0.226</a> .
  • L. Luperi Baglini, Partition regularity of nonlinear polynomials: a nonstandard approach, in: The Seventh European Conference on Combinatorics, Graph Theory and Applications (J. Nešetřil, M. Pellegrini, eds.), CRM Series, Vol. 16 (2013), 407–412, doi: 10.1007/978-88-7642-475-5_65.
  • L. Luperi Baglini, Hyperintegers and Nonstandard Techniques in Combinatorics of Numbers, PhD Dissertation (2012), University of Siena, arXiv: 1212.2049.
  • W.A.J. Luxemburg, A General Theory of Monads, in: Applications of Model Theory to Algebra, Analysis and Probability, (W. A. J. Luxemburg eds.), Holt, Rinehart, and Winston, New York (1969), 18–86.
  • J. Moreira, Monochromatic sums and products in $\mathbb{N}$, Ann. of Math., 185 (2017), 1069--1090.
  • J. Sahasrabudhe, Exponential Patterns in Arithmetic Ramsey Theory, arXiv:1607.08396.
  • J. Sahasrabudhe, Monochromatic Solutions to Systems of Exponential Equations, arXiv:1608.00109.
  • I. Schur, Uber die Kongruenz x^{m}+y^{m}=z^{m}(\mod p), Jahresber. Dtsch. Math.-Ver., Vol. 25 (1916), 114–117.
  • A. Sisto, Exponential Triples, Electron. J. Combin., Vol. 18 (2011), Paper 147.