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Reed's Math Ideas

Poly-chromatic and tuple-chromatic Ramsey Theory

I have been working on a generalization of Ramsey Theory. The concept is very simple for those who already know a little about Ramsey Theory. Rather than limit the research to sets having monochrome subsets with a given property, the generalization allows the subsets to be colored with specific groupings of colors. Check out the images made in working with type-chromatic Ramsey numbers: tuple-chromatic Ramsey Witnesses.

I wrote a paper on the subject: http://www.keldesign.com/math/TCRamsey/Tuple-Chromatic-Ramsey.pdf. In the paper, I attempt to make a mathematical definition for poly-chromatic and tuple-chromatic Ramsey numbers and poly-chromatic and tuple-chromatic Van der Waerden numbers. This is not so much a new theory, as it is an interesting variant of existing Ramsey Theory. In any case, I believe that the generalization provides many new areas of research. Many of the conjectures for existing Ramsey-type problems can have poly and tuple-chromatic counterparts.

One thing that is very appealing about tuple-chromatic Ramsey-type numbers is that they tend to be smaller then their classical Ramsey-type counterparts. This makes it more feasible to use brute-force to discover many of these numbers. Also, I am sure that many of the existing techniques for finding new bounds on Ramsey-type numbers will work well on poly/tuple-chromatic Ramsey-type numbers.

I wrote two programs for finding tuple-chromatic (tc) Ramsey and Van der Waerden numbers. Technically, the tc Ramsey program is only used to discover witnesses (lower bounds), whereas the tc Van der Waersen program can test all possibilities and produce an actual tc Van der Waerden number. The programs are found at: http://www.keldesign.com/math/TCRamsey/Code/find_min_comlete.zip, and http://www.keldesign.com/math/TCRamsey/Code/find_vdw.zip. They are not elegantly written. I wrote my own Vector class to expose the arrays as public. I know that this is wrong, but it improved the programs' speed significantly. If you have any trouble compiling or running these programs, please send me an email (math..at..keldesign.com).


Collatz, Fibonacci and Pascal

The following two examples are based on the concept of a Collatz sequence. You can find more information on Collatz sequences on Wikipedia: Collatz conjecture.

The first example is of a combination of the concepts of the Fibonacci sequence and the concept of Collatz sequences: Half-Fibonacci. This sequence is further described in The Database of Integer Sequences.

The second example is a combination of the concepts of Pascal's Triangle and Collatz Sequences. Check out my paper on Collatz-Pascal triangles here: CollatzPascal.


Mathematical Tricks

Summing digits trick

This trick is fun to do with someone who likes to do arithmetic.  It doesn't take that much effort and it can amaze an audience.  There are endless variations to this trick, but you need to understand a little modular arithmetic to make them up yourself.

  1. Ask for a volunteer who has pen and paper and doesn't mind doing some arithmetic (or has a calculator).
  2. Have the volunteer perform the following steps without showing you her/his work.
  3. Write down a three digit number with two requirements:
    • The three digits cannot be the same (e.g. 333 is not allowed)
    • The first digit is not the smallest one.
  4. Rearrange the digits so that the new number is less than the first.
  5. Subtract the second number from the first.
  6. Add 7 to the result.
  7. Double the result.
  8. Add up all the digits.
  9. Add up all the digits, again.
  10. The final result must be 5.  (you can say something like: "Dr. Magini thinks that your number is 5")

Variation

Just to show you that another variation is easy to concoct, here is another:

  1. Ask for a volunteer, as above.
  2. Have the volunteer perform the following steps without showing you her/his work.
  3. Write down a two digit number.
  4. Multiply the number by itself. (square the number)
  5. Rearrange the digits of the product so that the new number is less than the first.
  6. Subtract the rearranged number from the product.
  7. Add up all the digits.
  8. Add up all the digits, again.
  9. The final result must be 9.  (you can say: "The spirits are telling me to say 9")