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A very famous result in Ramsey Theory says that for many linear homogeneous equations there is a threshhold value $N(r)$ such that for any $r$-coloring of the integers in the interval $[1,\dots,n]$, with $n>N(r)$, there exist at least one monochromatic solution. This type of classical result was studied in the late 1800’s by Hilbert, Schur, and Rado before Ramsey generalized it.  One can further ask an enumeration question: what is the minimum number of monochromatic solutions possible in terms of $n$ over all r-colorings?    Several authors have estimated this function before (e.g., Frankl, Graham and R\”odl for regular equations), but there are a lot of open problems left. In this talk we discuss  new bounds on this problem using tools from Integer and Semi-definite Optimization that help find either optimal or near optimal 2-colorings minimizing the number of monochromatic solutions of several families of 3-variable non-regular homogeneous linear equations. In the last part we further extend our tools to 3 and 4 colorings of the famous Schur equation, improving earlier work.  This is joint work with Jesus De Loera, Amy Wang and William Wesley.