The Beauty of Combinatorics November 1, 2012 () The Beauty of Combinatorics November 1, 2012 1 / 19
Moshe Rosenfeld Institute of Technology University of Washington and Vietnam National University Hanoi University of Science () The Beauty of Combinatorics November 1, 2012 1 / 19
Moshe Rosenfeld Institute of Technology University of Washington and Vietnam National University Hanoi University of Science East China Normal University Shanghai, Nov. 1, 2012 () The Beauty of Combinatorics November 1, 2012 1 / 19
Introduction What is mathematics? 1 () The Beauty of Combinatorics November 1, 2012 2 / 19
Introduction What is mathematics? 1 "Mathematics is the study of numbers, shapes and 2 patterns." () The Beauty of Combinatorics November 1, 2012 2 / 19
Introduction What is mathematics? 1 "Mathematics is the study of numbers, shapes and 2 patterns." I also like the following quote by Sun-Yung Alice Chang 3 Professor of Mathematics, Princeton University: () The Beauty of Combinatorics November 1, 2012 2 / 19
Introduction What is mathematics? 1 "Mathematics is the study of numbers, shapes and 2 patterns." I also like the following quote by Sun-Yung Alice Chang 3 Professor of Mathematics, Princeton University: "Mathematics is a language like music. To learn it 4 systematically, it is necessary to master small pieces and gradually add another piece and then another. Mathematics is like the classical Chinese language - very polished and very elegant. Sitting in a good mathematics lecture is like sitting in a good opera. Everything comes together." () The Beauty of Combinatorics November 1, 2012 2 / 19
Introduction What is mathematics? 1 "Mathematics is the study of numbers, shapes and 2 patterns." I also like the following quote by Sun-Yung Alice Chang 3 Professor of Mathematics, Princeton University: "Mathematics is a language like music. To learn it 4 systematically, it is necessary to master small pieces and gradually add another piece and then another. Mathematics is like the classical Chinese language - very polished and very elegant. Sitting in a good mathematics lecture is like sitting in a good opera. Everything comes together." I am not planning to sing in this lecture... 5 () The Beauty of Combinatorics November 1, 2012 2 / 19
“Elementary problems, a sample” Not too long ago Graph Theory was dubbed as the The Slums of Topology combinatorics was treated as a curious side issue in mathematics. () The Beauty of Combinatorics November 1, 2012 3 / 19
“Elementary problems, a sample” Not too long ago Graph Theory was dubbed as the The Slums of Topology combinatorics was treated as a curious side issue in mathematics. Today combinatorics and Graph Theory are the most active research areas in Mathematics. () The Beauty of Combinatorics November 1, 2012 3 / 19
“Elementary problems, a sample” Not too long ago Graph Theory was dubbed as the The Slums of Topology combinatorics was treated as a curious side issue in mathematics. Today combinatorics and Graph Theory are the most active research areas in Mathematics. In this talk I will attempt to do justice to combinatorics. () The Beauty of Combinatorics November 1, 2012 3 / 19
“Elementary problems, a sample” Not too long ago Graph Theory was dubbed as the The Slums of Topology combinatorics was treated as a curious side issue in mathematics. Today combinatorics and Graph Theory are the most active research areas in Mathematics. In this talk I will attempt to do justice to combinatorics. Many combinatorial problems have an “elementary” flavor. They are easily understood even by high school students. I will discuss a sample of such problems, solutions and open problems. () The Beauty of Combinatorics November 1, 2012 3 / 19
“Elementary problems, a sample” Not too long ago Graph Theory was dubbed as the The Slums of Topology combinatorics was treated as a curious side issue in mathematics. Today combinatorics and Graph Theory are the most active research areas in Mathematics. In this talk I will attempt to do justice to combinatorics. Many combinatorial problems have an “elementary” flavor. They are easily understood even by high school students. I will discuss a sample of such problems, solutions and open problems. I will also try to demonstrate how combinatorics today actively interacts with many “classical” areas of mathematics. Let me start with a “topological” classic: () The Beauty of Combinatorics November 1, 2012 3 / 19
Brouwer’s fixed point theorem Theorem (Sperner) Let T 1 , T 2 , . . . T k be a triangulation of a triangle T . Assume that the vertices of T are colored red, blue and green. Further assume that on the green-red edge of T all points of the triangulation are colored green or red, on the green-blue edge all points are colored blue or green and on the red-blue edge all points are colored red or blue. Then there is a “rainbow” tiriangle T i . () The Beauty of Combinatorics November 1, 2012 4 / 19
Brouwer’s fixed point theorem Theorem (Sperner) Let T 1 , T 2 , . . . T k be a triangulation of a triangle T . Assume that the vertices of T are colored red, blue and green. Further assume that on the green-red edge of T all points of the triangulation are colored green or red, on the green-blue edge all points are colored blue or green and on the red-blue edge all points are colored red or blue. Then there is a “rainbow” tiriangle T i . Proof. () The Beauty of Combinatorics November 1, 2012 4 / 19
Brouwer’s fixed point theorem Theorem (Sperner) Let T 1 , T 2 , . . . T k be a triangulation of a triangle T . Assume that the vertices of T are colored red, blue and green. Further assume that on the green-red edge of T all points of the triangulation are colored green or red, on the green-blue edge all points are colored blue or green and on the red-blue edge all points are colored red or blue. Then there is a “rainbow” tiriangle T i . Proof. Count the number of rainbow edges in each triangle T i modulo 2. 1 () The Beauty of Combinatorics November 1, 2012 4 / 19
Brouwer’s fixed point theorem Theorem (Sperner) Let T 1 , T 2 , . . . T k be a triangulation of a triangle T . Assume that the vertices of T are colored red, blue and green. Further assume that on the green-red edge of T all points of the triangulation are colored green or red, on the green-blue edge all points are colored blue or green and on the red-blue edge all points are colored red or blue. Then there is a “rainbow” tiriangle T i . Proof. Count the number of rainbow edges in each triangle T i modulo 2. 1 Deduce that there is an odd number of Rainbow triangles. 2 () The Beauty of Combinatorics November 1, 2012 4 / 19
Brouwer fixed point theorem Theorem Let f : D − → D be a continuous function on the disc D. Then f has a fixed point. () The Beauty of Combinatorics November 1, 2012 5 / 19
Brouwer fixed point theorem Theorem Let f : D − → D be a continuous function on the disc D. Then f has a fixed point. Proof. () The Beauty of Combinatorics November 1, 2012 5 / 19
Brouwer fixed point theorem Theorem Let f : D − → D be a continuous function on the disc D. Then f has a fixed point. Proof. As the topologists tell us, we may assume that 1 D = { ( x 1 , x 2 , x 3 ) | x 1 + x 2 + x 3 = 1 , 1 ≥ x i ≥ 0 . } () The Beauty of Combinatorics November 1, 2012 5 / 19
Brouwer fixed point theorem Theorem Let f : D − → D be a continuous function on the disc D. Then f has a fixed point. Proof. As the topologists tell us, we may assume that 1 D = { ( x 1 , x 2 , x 3 ) | x 1 + x 2 + x 3 = 1 , 1 ≥ x i ≥ 0 . } Color ( 0 , 0 , 1 ) 1 , ( 0 , 1 , 0 ) 2 and ( 1 , 0 , 0 ) 3 . 2 () The Beauty of Combinatorics November 1, 2012 5 / 19
Brouwer fixed point theorem Theorem Let f : D − → D be a continuous function on the disc D. Then f has a fixed point. Proof. As the topologists tell us, we may assume that 1 D = { ( x 1 , x 2 , x 3 ) | x 1 + x 2 + x 3 = 1 , 1 ≥ x i ≥ 0 . } Color ( 0 , 0 , 1 ) 1 , ( 0 , 1 , 0 ) 2 and ( 1 , 0 , 0 ) 3 . 2 For every point u ∈ D , let φ ( u ) = i where i is the smallest index 3 of u − f ( u ) which is > 0. () The Beauty of Combinatorics November 1, 2012 5 / 19
Brouwer fixed point theorem Theorem Let f : D − → D be a continuous function on the disc D. Then f has a fixed point. Proof. As the topologists tell us, we may assume that 1 D = { ( x 1 , x 2 , x 3 ) | x 1 + x 2 + x 3 = 1 , 1 ≥ x i ≥ 0 . } Color ( 0 , 0 , 1 ) 1 , ( 0 , 1 , 0 ) 2 and ( 1 , 0 , 0 ) 3 . 2 For every point u ∈ D , let φ ( u ) = i where i is the smallest index 3 of u − f ( u ) which is > 0. () The Beauty of Combinatorics November 1, 2012 5 / 19
Brouwer fixed point theorem Theorem Let f : D − → D be a continuous function on the disc D. Then f has a fixed point. Proof. As the topologists tell us, we may assume that 1 D = { ( x 1 , x 2 , x 3 ) | x 1 + x 2 + x 3 = 1 , 1 ≥ x i ≥ 0 . } Color ( 0 , 0 , 1 ) 1 , ( 0 , 1 , 0 ) 2 and ( 1 , 0 , 0 ) 3 . 2 For every point u ∈ D , let φ ( u ) = i where i is the smallest index 3 of u − f ( u ) which is > 0. This is a “Sperner” coloring. 4 () The Beauty of Combinatorics November 1, 2012 5 / 19
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