Week 1 June 5 - June 11
Preliminary studies on Ramsey Theory: Ramsey's Theorem,
Van der Waerden's Theorem, and Schur's Theorem
Began researching Rado's Numbers, Coloring Finite Vector Spaces,
Completed Presentation. slides
Week 2 June 13 - June 17
Practice using Wolfram Mathematica and created basic for-loop algorithm to extrapolate numbers which do not belong to a color set C. This algorithm helped solve 2-Colors. Any number which is not red must be blue and vice versa. The equation being considered is ax+by=cz
Started working on an algorithm for 3-Colors which are far more elusive. Created basic NOT function, replacing the for-loop and 'patch' functions to remove infinity from sets. Implemented a while loop as well. Only not blue and not green numbers are red, and similarly for blue and green.
Week 3 June 20 - June 24
A cleaner version of the previous 3-Coloring was made. Researching recurvsive functions. The need for recursion was discovered when deducing values for all numbers proved to be impossible with out making choices.
A basic recursive algorithm or idea sketched out. Several recursive functions attempted, many errors caused by recursive depth errors.
In retrospect, the issues were due to using global variables instead of local ones.
Week 4 June 27 - July 1
First serious '3 Color Draft 1 " created. An attempt to make a nonrecursive function by coding decision making into mod 3 code. For an example:
Reads that the first choice was blue, then red, then green...
While attempting to make a recursive function that works, the nonrecursive model was cleaned up.
Week 5 July 5 - July 8
Happy Independence Day! A total start from scratch was made in order to create an error free recursive function defined by local variables. The code is significantly shorter than before.
Week 6 July 11 - July 15
"3 Color Draft 2" was created and is the first successful recursive function. Max[28, Null] issue resolved -- now only numbers are outputs. The recursive function extrapolates new numbers and then uses them all until a decision needs to be made. Then it makes 3 parallel decisions: color the smallest uncolored number Red,Blue, or Green and repeat.
With known solutions, sample equations are tested, numbers smaller than the real Rado numbers come out :-(.
Week 7 July 18 - July 22
BINGO. Code finally gives right answers after using Echo function to allow me to see 'under the hood'. Changes are made in which new numbers are colored (for a given color): only those new numbers which are smaller than the largest colored number M is considered or the smallest new number that is larger than M. Over determination was discovered -- coloring numbers larger than the Rado number that yield monochromatic solutions cause the code to prematurely stop and give the 'Rado number'
The code was made faster by changing a small detail:"only those new numbers which are smaller than the largest colored number M is considered AND the smallest new number that is larger than M." The timing for our most complicated solution R(2x+y=2z) went from above 12 hours and indefinite to 3.255 hours.
Week 8 July 24 - July 28
The code has been made more efficient and solves
x+2y=2z in 0.3 seconds with R = 14
x+2y=z in 33 seconds with R = 43
2x+3y=2z in 3.255 hours with R = 61
3x+y=z in 2.35 hours with R = 79
Several more equations will be attempted to be solve:
The final report has been written.
I am indebted to DIMACS (directors and staff) for the support and opportunity to have a math REU 'under my belt' for my career.
Last updated: July 28, 2016