# Kellen Myers - Homepage

Office: Hill 606, Office Hours: see here,
kellenm [at] math [dot] rutgers [dot] edu

Welcome to my homepage. I am a Mathematics PhD student at Rutgers University. My advisor is Doron Zeilberger. By the usual mathematicians' convention, this site is perpetually under construction. For now, feel free to read any of the papers below.

### Kellen Myers & Aaron Robertson, Two Color Off-Diagonal Rado-Type Numbers, Abstract:

We show that for any two linear homogeneous equations E0, E1, each with at least three variables and coefficients not all the same sign, any 2-coloring of Z+ admits monochromatic solutions of color 0 to E0 or monochromatic solutions of color 1 to E1. We define the 2-color off-diagonal Rado number RR(E0; E1) to be the smallest N such that [1,N] must admit such solutions. We determine a lower bound for RR(E0, E1) in certain cases when each Ei is of the form a1x1 + ... + anxn = z as well as find the exact value of RR(E0, E1) when each is of the form x1 + a2x2 + ... + anxn = z. We then present a Maple package that determines upper bounds for off-diagonal Rado numbers of a few particular types, and use it to quickly prove two previous results for diagonal Rado numbers.

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### Kellen Myers & Aaron Robertson, Two Color Off-Diagonal Rado-Type Numbers, Abstract:

There exists a minimum integer N such that any 2-coloring of {1, 2, ... , N} admits a monochromatic solution to x+y+kz = lw for k,l in Z+, where N depends on k and l. We determine N when l-k is in {0, 1, 2, 3, 4, 5}, and for all k, l for which (1/2)((l-k)2 -2)(l-k+1) ≤ k ≤ l - 4, as well as for arbitrary k when l = 2.

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Seminars I like to attend [click]:

My research interests are in combinatorics. The problems I work on are in the area of Ramsey theory, in particular Diophantine Ramsey theory, the study of the Ramsey-theoretic properties of the integer solutions to equations. My research focuses on computer-diven methods, but I am also interested in analytical methods in Diophantine Ramsey theory, which is to say additive combinatorics. I also enjoy problems and methods from other areas, like hypergraph theory and game theory. I would characterize my interest in mathematics, in general, as very broad. There are many topics of mathematics that I find very compelling, although I only have a basic familiarity with them. I enjoy the interdisciplinary nature of mathematics and appreciate collaboration between mathematicians with diverse interests. I am also a proponent of computer-based pedagogy and active learning in the mathematics classroom. Also, my Erdős number is 3 (check here).

K. Myers → A. Robertson → T. C. Brown → P. Erdős

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