Arithmetic progressions, quasi progressions, and Gallai-Ramsey colorings

We investigate several functions related to the r-color off-diagonal van der Waerden numbers w(m1,…,mr), where w(m1,…,mr) is the minimal integer n such that every r-coloring of {1,2,…,n} admits an mi-term arithmetic progression with all terms of color i for some i∈{1,2,…,r}. We start by giving a new lower bound for these related numbers. Next, the exact values and bounds of numbers related to quasi-progressions and mixed quasi-progression-van der Waerden numbers are given. Then, inspired by the success of graph Gallai-Ramsey theory and rainbow arithmetic progressions, we introduce the concept of Gallai-van der Waerden numbers, and obtain some exact values and bounds for these numbers, some of which are derived by the probabilistic method and the Lovász Local Lemma.