Deriving Ramsey Numbers

1. Deriving Ramsey Numbers Interim Project Evaluations 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 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) Deriving Ramsey Numbers: Team 7, Albuquerque Academy

4. Definitions 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 Possible to create a mapping without a k3 in a 5-node graph (a k5) Deriving Ramsey Numbers: Team 7, Albuquerque Academy

5. Problem Statement • Create program to find Ramsey number K(r,b) for given r and b 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 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 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 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 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. 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 Deriving Ramsey Numbers: Team 7, Albuquerque Academy

12. Questions