We investigate several functions related to the r-color off-diagonal van der Waerden
numbers w (m 1,…, mr), where w (m 1,…, mr) is the minimal integer n such that every r …

The centralized coded caching focuses on reducing the network burden in peak times in a
wireless network system. In this paper, motivated by the study of the only rainbow 3-term …

Several network communication problems are highly related such as coded caching and
distributed computation. The centralized coded caching focuses on reducing the network …

In this paper, we study the rainbow Erd\H {o} s-Rothschild problem with respect to 3-term
arithmetic progressions. We obtain the asymptotic number of $ r $-colorings of $[n] $ without …

By recalling van der Waerden theorem, there exists a least a positive integer w= w (k; r) such
that for any n≥ w, every r-colouring of [1, n] admits a monochromatic k-term arithmetic …

For a finite commutative ring R, let a, b, c∈ R be fixed elements. Consider the equation ax+
by= cz where x, y, and z are idempotents, units, and any element in the ring R, respectively …

We study graph theory and combinatorics which are topics in discrete mathematics. The
graphs we consider in the thesis consist of a set of vertices and a set of edges in which every …