Combinatorial Graph Theory
Exercise Sheet 11
Prof. Dr. Ir. Gerhard Woeginger WS 2016/17
Tim Hartmann Due Date: January 30, 16:15
Department of Computer Science RWTH Aachen University
• Hand in your solutions in a group of two or three students.
• Write the name and enrollment number of each group member on every sheet that you hand in.
• To achieve the permission for the exam you must earn 50% of the sum of all points.
• You can earn 50% bonus points by presenting your solution. At the beginning of every exercise session, you can mark the exercises that you want to present.
• If a student is not able to present a correct solution although he/she marked the exercise as presentable, he/she will lose all of his/her points on the exercise sheet.
Exercise 1 (1+3 points)
This exercise asks you to establishR(3,5) = 14.
(a) Show R(3,5)≤14 by combining some of the facts that you saw in class.
(b) For showingR(3,5)>13, consider a red-blue coloring of the edges ofK13on vertex set {1,2, . . . ,13}.
Hint: You may want to color red all the edges[i, i+1]and all the edges[i, i+x], wherexis an appropriately chosen value with 2≤x≤6.
Exercise 2 (3 points)
Show that a triangle-free graph on nvertices has an independent set of size Ω(√ n).
Hint: Consider the neighborhood of some fixed vertex.
Exercise 3 (3 points)
Prove that for every k, there exists an integer N = N(k) with the following property: If the subsets of {1,2, . . . , N}arek-colored, then there are two disjoint non-empty subsetsX andY, such that the three sets X,Y, and X∪Y all have the same color.
Hint: For coloring the edge [i, j] in an underlying complete graph KN, use the color of an appropriately chosen interval in{1,2, . . . , N}.
Exercise 4 (3+3 points)
• (a) Construct a 2-coloring of {1,2, . . . ,2055} so that there is no mono-chromatic solution to the equation 8x+ 8y=z.
Hint: Let the blue numbers form an interval.
• (b) Show that under any 2-coloring of{1,2, . . . ,2056}, there exists a mono-chromatic solution to the equation 8x+ 8y=z.
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