- W.J. Wesley,
*Improved Ramsey-type theorems for Fibonacci numbers and other sequences***Abstract:**Van der Waerden's theorem states that for any positive integers*k*and*r*, there exists a smallest value*n=w(k,r)*, called the van der Waerden number, such that every*r*-coloring of*{1,…,n}*contains a monochromatic*k*-term arithmetic progression. We consider two variants of van der Waerden numbers: the numbers*n=n(AP*, the smallest value where every_{D},k;r)*r*-coloring of*{1,…,n}*contains a monochromatic*k*-term arithmetic progression with common difference in*D*, and the numbers*n=Δ(D,k;r)*, the smallest value*n*where every*r*-coloring of*{1,…,n}*contains a sequence*x*where the differences between consecutive terms are members of_{1},...,x_{k}*D*. We study the case when*D*is set of Fibonacci numbers*F*and give improved bounds for the largest*r*where*n(AP*and_{F},k;r)*Δ(F,k;r)*exist for all*k*. Moreover, we give some computational data on*Δ(D,k;r)*for other sets*D*. - J.A. De Loera and W.J. Wesley,
*Ramsey Numbers through the Lenses of Polynomial Ideals and Nullstellensätze***Abstract:**In this article we study the Ramsey numbers*R(r,s)*through Hilbert's Nullstellensatz and Alon's Combinatorial Nullstellensatz. We give polynomial encodings whose solutions correspond to Ramsey graphs of order*n*, those that do not contain a copy of K_{r}or K_{s}. When these systems have no solution and*n=R(r,s)*, we construct Nullstellensatz certificates whose degrees are equal to the restricted online Ramsey numbers introduced by Conlon, Fox, Grinshpun and He. Moreover, we show that these results generalize to other numbers in Ramsey theory, including Rado, van der Waerden, and Hales-Jewett numbers. Finally, we introduce a family of numbers that relate to the coefficients of a certain "Ramsey polynomial" that gives lower bounds for Ramsey numbers. - Y. Chang, J.A. De Loera, and W.J. Wesley,
*Rado numbers and SAT computations*, Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation (2022).**Abstract:**Given a linear equation*E*, the*k*-color Rado number*R*is the smallest integer n such that every_{k}(E)*k*-coloring of*{1,2,3,...,n}*contains a monochromatic solution to*E*. The degree of regularity of*E*, denoted*dor(E)*, is the largest value*k*such that*R*is finite. In this article we present new theoretical and computational results about the Rado numbers_{k}(E)*R*and the degree of regularity of three-variable equations_{3}(E)*E*. -
M. Barnett, A. Folsom, and W.J. Wesley,
*Rank generating functions for odd-balanced unimodal sequences*, Journal of the Australian Mathematical Society 109(2) (2020).**Abstract:**Let*μ(m,n)*(respectively,*η(m,n)*) denote the number of odd-balanced unimodal sequences of size 2*n*and rank*m*with even parts congruent to 2 mod 4 (respectively, 0 mod 4 ) and odd parts at most half the peak. We prove that two-variable generating functions for*μ(m,n)*and*η(m,n)*are simultaneously quantum Jacobi forms and mock Jacobi forms. These odd-balanced unimodal rank generating functions are also duals to partial theta functions originally studied by Ramanujan. Our results also show that there is a single**C**^{∞}function in**R**×**R**to which the errors to modularity of these two different functions extend. We also exploit the quantum Jacobi properties of these generating functions to show, when viewed as functions of the two variables*w*and*q*, how they can be expressed as the same simple Laurent polynomial when evaluated at pairs of roots of unity. Finally, we make a conjecture which fully characterizes the parity of the number of odd-balanced unimodal sequences of size 2*n*with even parts congruent to 0 mod 4 and odd parts at most half the peak. -
M. Barnett, A. Folsom, O. Ukogu, W.J. Wesley, and H. Xu,
*Quantum Jacobi forms and balanced unimodal sequences*, Journal of Number Theory 186 (2018).**Abstract:**The notion of a quantum Jacobi form was defined in 2016 by Bringmann and the second author, marrying Zagier's notion of a quantum modular form [12] with that of a Jacobi form. Only one example of such a function has been given to-date. Here, we prove that two combinatorial rank generating functions for certain balanced unimodal sequences, studied previously by Kim, Lim and Lovejoy, are also natural examples of quantum Jacobi forms. These two combinatorial functions are also duals to partial theta functions studied by Ramanujan. Additionally, we prove that these two functions have the stronger property that they exhibit mock Jacobi transformations in**C**×**H**as well as quantum Jacobi transformations in**Q**×**Q**. As corollaries to these results, we use quantum Jacobi properties to establish new, simpler expressions for these functions as simple Laurent polynomials when evaluated at pairs of rational numbers.