Ramsey Theory 2

2/7/2017 Ma/CS 6b Class 13: Ramsey Theory 2 Frank P. Ramsey By Adam Sheffer Ramsey Numbers (Special Case) 푝1, … , 푝푘 – positive integers. The Ramsey number 푅 푝1, … , 푝푘; 2 is the minimum integer 푛 such that for every coloring of the edges of 퐾푛 using 푘 colors, at least one of the following subgraphs exists: ◦ A 퐾푝1 colored only with color 1. ◦ A 퐾푝2 colored only with color 2. ◦ … ◦ A 퐾푝푘 colored only with color 푘. 1 2/7/2017 An Example The expression 푛 = 푅 3,3,3,3; 2 means that every coloring of the edges of 퐾푛 using four colors contains a monochromatic triangle. Recall: 푅(3,3) We already proved that 푅 3,3; 2 = 6. Any red-blue coloring 퐾6 contains a monochromatic triangle. ◦ This is not the case for 퐾5. 2 2/7/2017 Recall: 푅(2,4) What is 푅(2,4)? ◦ Either there is a red 퐾2, which is just a red edge, or there is a blue 퐾4. ◦ This is the case for any coloring of 퐾4, but not for any coloring of 퐾3. ◦ Thus, 푅(2,4) = 4. Homogenous Subsets 푆 – a set of elements (e.g., vertices). 푆 ◦ For an integer 1 ≤ 푟 ≤ 푆 , we let the set 푟 of all subsets of 푟 elements of 푆. 푆 ◦ We give every subset of a color. 푟 ◦ We say that 푆′ ⊂ 푆 is 푖-homogeneous if every 푆′ subset of is of color 푖. 푟 3 2/7/2017 Example: Subsets of Size 2 When the subsets are of size two, this is equivalent to coloring edges of a complete graph. Subsets of Size 3 If 푟 = 3, we color triples of vertices. ◦ Can be thought of as coloring the triangular faces of 퐾푛. 4 2/7/2017 General Ramsey Numbers 푟, 푝1, … , 푝푘 – positive integers. The Ramsey number 푅 푝1, … , 푝푘; 푟 is the minimum integer 푛 such that every 1,2, … , 푛 coloring of using 푘 colors 푟 yields an 푖-homogeneous set of size 푝푖, for some 1 ≤ 푖 ≤ 푘. A Bit of Intuition If 푟 = 3, we color triples of vertices. ◦ Can be thought of as coloring the triangular faces of 퐾푛. ◦ We are looking for a large subset 푆 of the vertices, such that each triangle that is spanned by three vertices of 푆 has the same color. Monochromatic 퐾4 5 2/7/2017 The Case of Larger 푟 We already know that 푅 푝1, 푝2; 2 is finite for any positive 푝1 and 푝2. ◦ We now generalize this to the case of larger 푟. Theorem. For any positive integers 푟, 푝1, 푝2, the Ramsey number 푅 푝1, 푝2; 푟 is finite. What Do We Need to Prove? For any positive integers 푟, 푝1, 푝2, we need to prove that there exists 푛 such that every coloring of the 푟-tuples of a set of size 푛 in red and blue contains: ◦ Either a subset of 푝1 elements such that every 푟-tuple in it is red, ◦ or a subset of 푝2 elements such that every 푟-tuple in it is blue. 6 2/7/2017 Proof We use a double induction. ◦ We prove the theorem by induction on 푟. ◦ We prove the induction step using an induction on 푝1 + 푝2. Induction basis (on 푟). ◦ When 푟 = 1, we can simply take 푝1 + 푝2 − 1 elements. ◦ We already proved the case of 푟 = 2. Induction Step We prove the induction step for a given value of 푟 by induction on 푝1 + 푝2. ◦ Induction basis. If 푝1 < 푟 or 푝2 < 푟 then the claim vacuously holds for sets of 푝푖 elements, since they have zero 푟-tuples. ◦ Induction step. We set 푞1 = 푅 푝1 − 1, 푝2; 푟 , 푞2 = 푅 푝1, 푝2 − 1; 푟 , and 푛 = 1 + 푅 푞1, 푞2; 푟 − 1 . ◦ By the hypotheses, 푞1, 푞2, and 푛 are finite. 7 2/7/2017 Proof (cont.) 푞1 = 푅 푝1 − 1, 푝2; 푟 , 푞2 = 푅 푝1, 푝2 − 1 , 푛 = 1 + 푅 푞1, 푞2, 푟 − 1 . Let 푆 be a set of 푛 elements, with every 푟-tuple colored either red or blue. ◦ Pick an element 푥 ∈ 푆 and let 푆′ = 푆 ∖ {푥}. ◦ We color an 푟 − 1 -tuple 푇 ⊂ 푆′ using the same color as the 푟-tuple 푇 ∪ 푥 . ′ ◦ Since 푆 = 푛 − 1 = 푅 푞1, 푞2, 푟 − 1 , there is either a red subset of 푞1 elements (with respect to the colors of the 푟 − 1 -tuples), or a blue subset of 푞2 elements. Completing the Proof 푞1 = 푅 푝1 − 1, 푝2; 푟 , 푞2 = 푅 푝1, 푝2 − 1 , 푛 = 1 + 푅 푞1, 푞2, 푟 − 1 . 푆 – a set of 푛 elements. 푆′ = 푆 ∖ 푥 . WLOG, we assume that there is a red subset 푆푟 of 푞1 elements (with respect to 푟 − 1 -tuples). We consider the 푟-tuples of 푆푟. Since 푆푟 = 푞1 = 푅 푝1 − 1, 푝2; 푟 , either there is a blue subset of size 푝2, or a red subset of size 푝1 − 1. In the latter case, by adding 푥 to the subset, we obtain a red subset of size 푝1. 8 2/7/2017 Ramsey’s Theorem A straightforward extension of the proof yields a more general result. Theorem. For any positive integers 푟, 푝1, … , 푝푘, the Ramsey number 푅 푝1, … , 푝푘; 푟 is finite. ◦ Intuitively: In every sufficiently large arbitrary object (i.e., an arbitrary coloring) there must be some order (i.e., a monochromatic subset). Some History In the early 1930’s in Budapest, a few students used to regularly meet on Sundays in a specific city park bench. Among the participants were Paul Erdős, George Szekeres, and Esther Klein. Klein told the group that for any set of five points with no three on a line, four of the points are the vertices of a convex quadrilateral. 9 2/7/2017 The convex hull of a point set 푆 is the smallest convex polygon that contains 푆. Given a set of five points: ◦ If the convex hall of the point set has at least four points in it, then we are done. ◦ If the convex hull consists of three points, we consider the line ℓ that passes through the two interior points. ◦ We take the two interior points and the two points that are on the same side of ℓ. The Story Continues The Budapest students started to think about whether there exists an 푛 such that every set of 푛 points with no three on a line contains the vertices of a convex pentagon. ◦ More generally, does a similar condition hold for every convex 푘-gon? 10 2/7/2017 The Happy Ending Problem Even though Erdős was part of the group, the first to prove the claim was Szekeres. A couple of years later, Esther Klein, who suggested the problem married George Szekeres who solved it. ◦ Since then, this problem is called “the happy ending problem”. ◦ They lived together up to their 90’s. The Happy Ending Theorem Theorem. For every integer 푚 ≥ 3, there exists an integer 푁(푚) such that any set of 푁(푚) points with no three on a line contains a subset of 푚 points that are the vertices of a convex 푚-gon. 11 2/7/2017 First Claims Straightforward claim. Any four vertices of a convex 푛-gon span a convex quadrilateral. Less straightforward claim. If every four vertices of an 푛-gon 푃 form a convex quadrilateral, then 푃 is convex. Proving the Claim Claim. If every four vertices of an 푛-gon 푃 form a convex quadrilateral, then 푃 is convex. Proof. Assume for contradiction that 푃 is not convex. ◦ There is a vertex 푣 that is not in the convex hull of the vertices of 푃. ◦ Triangulate the convex hull of 푃. ◦ 푣 together with the three vertices 푣 of the triangle containing 푣 do not span a convex quadrelateral! 12 2/7/2017 Proving the Theorem Set 푁 푚 = 푅 푚, 5; 4 . ◦ Given a set of 푁 푚 points with no three on a line, we color every 4-tuple of points. ◦ A 4-tuple is colored red if it spans a convex quadrilateral, and otherwise blue. ◦ By Ramsey’s theorem, either there is a red subset of size 푚 and or blue one of size 5. ◦ But we proved that for any 5 points there must be a convex (=red) quadruple! ◦ Thus, there is a red subset of size 푚, and by the previous claim it spans a convex 푚-gon. The Erdős-Szekeres Conjecture In the 1930’s, Erdős and Szekeres proved 2푚 − 4 2푚−2 + 1 ≤ 푁 푚 ≤ + 1. 푚 − 2 Conjecture (Erdős and Szekeres). 푁 푚 = 2푚−2 + 1. Hardly any progress until 2016. 2/3 ◦ Suk proved 푁 푚 ≤ 2푚+6푚 log 푚. 13 2/7/2017 Frank P. Ramsey Died in 1930 at the age of 26. By then, he: ◦ Wrote mathematical papers, including getting a whole subfield named after him. ◦ Wrote several philosophical works. Wittgenstein mentions him in the introduction to his Philosophical Investigations as an influence. ◦ Wrote several economics papers, as a student to John Maynard Keynes. ◦ Had a wife, kids, etc. Graph Ramsey Theory So far we looked for monochromatic copies of some 퐾푚 in colorings of 퐾푛. ◦ We now move to searching monochromatic copies of other types of graphs. ◦ Given two graphs 퐺1, 퐺2, we denote by 푅 퐺1, 퐺2 the minimum number 푛 such that every coloring of 퐾푛 contains either a blue copy of 퐺1 or a red copy of 퐺2. Example. ◦ 푃푚 – a graph that is a path of length 푚.

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