Deriving Ramsey Numbers Interim Project Evaluations Team 7 Albuquerque Academy

February 24, 2007Deriving Ramsey Numbers: Team 7, Albuquerque Academy2 Contents Definitions Problem Statement Method and Mathematical Model Explored Algorithm Possibilities Expected Results Current Overall Progress Plans for the Future Questions

February 24, 2007Deriving Ramsey Numbers: Team 7, Albuquerque Academy3 Definitions Ramsey numbers part of mathematical field of graph theory k m 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 k r or k b )

February 24, 2007Deriving Ramsey Numbers: Team 7, Albuquerque Academy4 Definitions Possible to create a mapping without a k 3 in a 5-node graph (a k 5 ) Example: R(3,3) = 6 Not possible to create a coloring without a k 3 in a 6- node graph (a k 6 ) and this R(3,3) = 6

February 24, 2007Deriving Ramsey Numbers: Team 7, Albuquerque Academy5 Problem Statement Create program to find Ramsey number K(r,b) for given r and b

February 24, 2007Deriving Ramsey Numbers: Team 7, Albuquerque Academy6 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, 2007Deriving Ramsey Numbers: Team 7, Albuquerque Academy7 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, 2007Deriving Ramsey Numbers: Team 7, Albuquerque Academy8 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, 2007Deriving Ramsey Numbers: Team 7, Albuquerque Academy9 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, 2007Deriving Ramsey Numbers: Team 7, Albuquerque Academy10 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, 2007Deriving Ramsey Numbers: Team 7, Albuquerque Academy11 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

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