Deriving Ramsey NumbersInterim Project Evaluations

Team 7

Albuquerque Academy

February 24, 2007 Deriving Ramsey Numbers: Team 7, Albuquerque Academy 2

Contents• Definitions• Problem Statement• Method and Mathematical Model• Explored Algorithm Possibilities• Expected Results• Current Overall Progress• Plans for the Future• Questions

February 24, 2007 Deriving Ramsey Numbers: Team 7, Albuquerque Academy 3

Definitions• Ramsey numbers part of mathematical field of graph

theory

• km is defined as a graph containing m nodes and all possible line between the nodes

• Ramsey functions notated as K(r, b)=n– K is Ramsey function– r, b are independent variables– n is result of Ramsey function; called Ramsey number

• Ramsey function gives smallest graph size that when colored in any pattern of only two colors, will not contain sub-graphs of size r or b (i.e. does not contain a kr or kb)

February 24, 2007 Deriving Ramsey Numbers: Team 7, Albuquerque Academy 4

Definitions

Possible to create a mapping without a k3 in a 5-node graph (a k5)

Example: R(3,3) = 6

Not possible to create a coloring without a k3 in a 6-node graph (a k6) and this R(3,3) = 6

February 24, 2007 Deriving Ramsey Numbers: Team 7, Albuquerque Academy 5

Problem Statement• Create program to find Ramsey number K(r,b)

for given r and b

February 24, 2007 Deriving Ramsey Numbers: Team 7, Albuquerque Academy 6

Method and Mathematical Model• Begin at starting graph

• See if this graph fits the parameters of the inputs (m and n)

• Systematically recolor the graph and repeat

• Continue doing this until all graphs are tested

• If no graphs succeed, the answer is found

• Otherwise try a graph with one more node

February 24, 2007 Deriving Ramsey Numbers: Team 7, Albuquerque Academy 7

Explored Algorithm Possibilities• Eliminate some unnecessary graphs

• Work from graphs that are known to work with one less node

• Only try a few to see if those graphs can be eliminated by:

• Symmetry

• Correcting problem areas

February 24, 2007 Deriving Ramsey Numbers: Team 7, Albuquerque Academy 8

Expected Results• Successfully calculate Ramsey numbers for

small values of inputs r, b

• Make improvements over “brute force” base-speed in computing time required to calculate Ramsey numbers

• Still require significant amounts of computing time and power to calculate Ramsey numbers for larger inputs of r, b, as modern programs have encountered

February 24, 2007 Deriving Ramsey Numbers: Team 7, Albuquerque Academy 9

Current Overall Progress• Have completed thorough research phase

– Found information on Ramsey number research and discovered solutions

– Found mathematical theorems to assist in developing algorithm and reducing neccesary computing time and power

– Found basic information on past attempts at creating a program to calculate Ramsey numbers

February 24, 2007 Deriving Ramsey Numbers: Team 7, Albuquerque Academy 10

Current Overall Progress• Have basics of code for first version of program

– Create map using Hash Map dynamic data structure in Java:HashMap Frank = new HashMap(n); //Frank (Ramsey's first name) is THE map

– Query the user for the inputs:String temp;temp = JOptionPane.showInputDialog(null, "How many points would you like in the map?",“Deriving Ramsey Numbers :: Input", JOptionPane.QUESTION_MESSAGE );

– Will likely use objects to represent the line segments to create consistency when re-coloring.

February 24, 2007 Deriving Ramsey Numbers: Team 7, Albuquerque Academy 11

Plans for the Future• Continue writing program

– Initially, work with a “brute force” algorithm– Later, implement our more advanced and time

saving algorithm

• Continue to develop the algorithm based on experience while writing program

• Prepare final report summarizing our research

Questions