Unavoidable patterns in words
|
|
|
- Beatrice Gallagher
- 5 years ago
- Views:
Transcription
1 Unavoidable patterns in words Benny Sudakov ETH, Zurich joint with D.Conlon and J. Fox
2 Ramsey numbers Definition: The Ramsey number r k (n) is the minimum N such that every 2-coloring of the k-tuples of an N-element set contains a monochromatic set of order n.
3 Ramsey numbers Definition: The Ramsey number r k (n) is the minimum N such that every 2-coloring of the k-tuples of an N-element set contains a monochromatic set of order n. Theorem: (Ramsey 1930) For all k, n, the Ramsey number r k (n) is finite.
4 Ramsey numbers Definition: The Ramsey number r k (n) is the minimum N such that every 2-coloring of the k-tuples of an N-element set contains a monochromatic set of order n. Theorem: (Ramsey 1930) For all k, n, the Ramsey number r k (n) is finite. Question: Estimate the growth rate of r k (n).
5 Bounds on Ramsey numbers Theorem: 2 n/2 r 2 (n) 2 2n. (Erdős 47, Erdős Szekeres 35)
6 Bounds on Ramsey numbers Theorem: 2 n/2 r 2 (n) 2 2n. (Erdős 47, Erdős Szekeres 35) 2 cn2 r 3 (n) 2 2c n. (Erdős Rado 52, Erdős Hajnal 60s)
7 Bounds on Ramsey numbers Theorem: 2 n/2 r 2 (n) 2 2n. (Erdős 47, Erdős Szekeres 35) 2 cn2 r 3 (n) 2 2c n. (Erdős Rado 52, Erdős Hajnal 60s) Remarks: There is a similar gap of one exponential between the upper and the lower bound for r k (n) for k > 3. These bounds are towers of exponentials of height k and k 1 respectively.
8 Bounds on Ramsey numbers Theorem: 2 n/2 r 2 (n) 2 2n. (Erdős 47, Erdős Szekeres 35) 2 cn2 r 3 (n) 2 2c n. (Erdős Rado 52, Erdős Hajnal 60s) Remarks: There is a similar gap of one exponential between the upper and the lower bound for r k (n) for k > 3. These bounds are towers of exponentials of height k and k 1 respectively. Determining the behavior of r 3 (n) will close the gap for all k due to stepping-up lemma of Erdős Hajnal, which constructs lower bound colorings for uniformity k + 1 from colorings for uniformity k, effectively gaining an extra exponential each time it is applied.
9 Words and patterns Definition Words and patterns are strings of characters over fixed alphabets.
10 Words and patterns Definition Words and patterns are strings of characters over fixed alphabets. A subword of a word is a block of consecutive letters.
11 Words and patterns Definition Words and patterns are strings of characters over fixed alphabets. A subword of a word is a block of consecutive letters. A word w contains the pattern P if there is a way to substitute a nonempty word for each letter in P so that the resulting word is a subword of w.
12 Words and patterns Definition Words and patterns are strings of characters over fixed alphabets. A subword of a word is a block of consecutive letters. A word w contains the pattern P if there is a way to substitute a nonempty word for each letter in P so that the resulting word is a subword of w. Example: The word mathematics contains the pattern xyxz with x = mat, y = he and z = ics.
13 Words and patterns Definition Words and patterns are strings of characters over fixed alphabets. A subword of a word is a block of consecutive letters. A word w contains the pattern P if there is a way to substitute a nonempty word for each letter in P so that the resulting word is a subword of w. Example: The word mathematics contains the pattern xyxz with x = mat, y = he and z = ics.
14 q-unavoidability Definition: A pattern P is q-unavoidable if every sufficiently long word over an alphabet of size q contains a copy of P.
15 q-unavoidability Definition: A pattern P is q-unavoidable if every sufficiently long word over an alphabet of size q contains a copy of P. Examples: Thue 1906: The pattern xx is 2-unavoidable, but 3-avoidable.
16 q-unavoidability Definition: A pattern P is q-unavoidable if every sufficiently long word over an alphabet of size q contains a copy of P. Examples: Thue 1906: The pattern xx is 2-unavoidable, but 3-avoidable. Thue 1912, Morse 1921: The pattern xxx is 1-unavoidable, but 2-avoidable.
17 q-unavoidability Definition: A pattern P is q-unavoidable if every sufficiently long word over an alphabet of size q contains a copy of P. Examples: Thue 1906: The pattern xx is 2-unavoidable, but 3-avoidable. Thue 1912, Morse 1921: The pattern xxx is 1-unavoidable, but 2-avoidable. Start with a and recursively substitute a ab and b ba.
18 q-unavoidability Definition: A pattern P is q-unavoidable if every sufficiently long word over an alphabet of size q contains a copy of P. Examples and applications of pattern avoidance: Combinatorics Group theory, e.g, Burnside problem, Undecidability Symbolic Dynamics Number theory
19 Unavoidability Definition: A pattern P is unavoidable if it is q-unavoidable for all q 1.
20 Unavoidability Definition: A pattern P is unavoidable if it is q-unavoidable for all q 1. Ramsey question for patterns: Which patterns are unavoidable?
21 Unavoidability Definition: A pattern P is unavoidable if it is q-unavoidable for all q 1. Ramsey question for patterns: Which patterns are unavoidable? Theorem: (Bean Ehrenfeucht McNulty 1979, Zimin 1984) A word is unavoidable if and only if it is contained in a Zimin word, defined recursively by Z 1 = x 1 and Z n = Z n 1 x n Z n 1.
22 Unavoidability Definition: A pattern P is unavoidable if it is q-unavoidable for all q 1. Ramsey question for patterns: Which patterns are unavoidable? Theorem: (Bean Ehrenfeucht McNulty 1979, Zimin 1984) A word is unavoidable if and only if it is contained in a Zimin word, defined recursively by Z 1 = x 1 and Z n = Z n 1 x n Z n 1. Zimin words: Z 1 = x
23 Unavoidability Definition: A pattern P is unavoidable if it is q-unavoidable for all q 1. Ramsey question for patterns: Which patterns are unavoidable? Theorem: (Bean Ehrenfeucht McNulty 1979, Zimin 1984) A word is unavoidable if and only if it is contained in a Zimin word, defined recursively by Z 1 = x 1 and Z n = Z n 1 x n Z n 1. Zimin words: Z 1 = x Z 2 = xyx
24 Unavoidability Definition: A pattern P is unavoidable if it is q-unavoidable for all q 1. Ramsey question for patterns: Which patterns are unavoidable? Theorem: (Bean Ehrenfeucht McNulty 1979, Zimin 1984) A word is unavoidable if and only if it is contained in a Zimin word, defined recursively by Z 1 = x 1 and Z n = Z n 1 x n Z n 1. Zimin words: Z 1 = x Z 2 = xyx Z 3 = xyxzxyx
25 Unavoidability Definition: A pattern P is unavoidable if it is q-unavoidable for all q 1. Ramsey question for patterns: Which patterns are unavoidable? Theorem: (Bean Ehrenfeucht McNulty 1979, Zimin 1984) A word is unavoidable if and only if it is contained in a Zimin word, defined recursively by Z 1 = x 1 and Z n = Z n 1 x n Z n 1. Zimin words: Z 1 = x Z 2 = xyx Z 3 = xyxzxyx Z 4 = xyxzxyxwxyxzxyx
26 Ramsey numbers for patterns Definition: Let f (n, q) be the smallest natural number such that any word of length f (n, q) over an alphabet of size q contains Zimin word Z n.
27 Ramsey numbers for patterns Definition: Let f (n, q) be the smallest natural number such that any word of length f (n, q) over an alphabet of size q contains Zimin word Z n. Problem: Estimate the asymptotics of f (n, q).
28 Ramsey numbers for patterns Definition: Let f (n, q) be the smallest natural number such that any word of length f (n, q) over an alphabet of size q contains Zimin word Z n. Problem: Estimate the asymptotics of f (n, q). Few upper bounds: f (1, q) = 1, follows easily from Z 1 = x.
29 Ramsey numbers for patterns Definition: Let f (n, q) be the smallest natural number such that any word of length f (n, q) over an alphabet of size q contains Zimin word Z n. Problem: Estimate the asymptotics of f (n, q). Few upper bounds: f (1, q) = 1, follows easily from Z 1 = x. f (2, q) = 2q + 1, follows easily from Z 2 = xyx.
30 Ramsey numbers for patterns Definition: Let f (n, q) be the smallest natural number such that any word of length f (n, q) over an alphabet of size q contains Zimin word Z n. Problem: Estimate the asymptotics of f (n, q). Few upper bounds: f (1, q) = 1, follows easily from Z 1 = x. f (2, q) = 2q + 1, follows easily from Z 2 = xyx. f (3, q) q q (Rytter Shur)
31 General upper bound Lemma: (Cooper Rorabaugh) f (n + 1, q) (f (n, q) + 1)(q f (n,q) + 1) 1
32 General upper bound Lemma: (Cooper Rorabaugh) f (n + 1, q) (f (n, q) + 1)(q f (n,q) + 1) 1 Proof: Given some word w, split it into m = q f (n,q) + 1 words w i of length f (n, q), which are separated by single letters: w 1 x w 2 y... z w m Then there are two identical words w i and w j, each containing the same copy of some Zimin word Z n. This forms Z n+1.
33 General upper bound Lemma: (Cooper Rorabaugh) f (n + 1, q) (f (n, q) + 1)(q f (n,q) + 1) 1 Proof: Given some word w, split it into m = q f (n,q) + 1 words w i of length f (n, q), which are separated by single letters: w 1 x w 2 y... z w m Then there are two identical words w i and w j, each containing the same copy of some Zimin word Z n. This forms Z n+1. Theorem: f (n, q) q q } n-1 times.
34 General upper bound Lemma: (Cooper Rorabaugh) f (n + 1, q) (f (n, q) + 1)(q f (n,q) + 1) 1 Proof: Given some word w, split it into m = q f (n,q) + 1 words w i of length f (n, q), which are separated by single letters: w 1 x w 2 y... z w m Then there are two identical words w i and w j, each containing the same copy of some Zimin word Z n. This forms Z n+1. Theorem: f (n, q) q q } n-1 times. Proof: Apply lemma recursively, starting with f (3, q) q q.
35 Lower bounds? Lemma: (Cooper Rorabaugh) f (n, q) q 2n 1 (1+o(1)), where the o(1) term depends on both q and n.
36 Lower bounds? Lemma: (Cooper Rorabaugh) f (n, q) q 2n 1 (1+o(1)), where the o(1) term depends on both q and n. Remark: Note that for any fixed n this bound is only constant degree polynomial in q versus upper bound which is tower-type!
37 Lower bounds? Lemma: (Cooper Rorabaugh) f (n, q) q 2n 1 (1+o(1)), where the o(1) term depends on both q and n. Remark: Note that for any fixed n this bound is only constant degree polynomial in q versus upper bound which is tower-type! Question: What is the maximum length of a word not containing the n-th Zimin word?
38 Tight lower bounds Theorem: (Conlon Fox S.) For fixed n and large q, f (n, q) q q o(q) } n-1 times.
39 Tight lower bounds Theorem: (Conlon Fox S.) For fixed n and large q, f (n, q) q q o(q) } n-1 times. f (n, 2) 2 2 } n-4 times.
40 Tight lower bounds Theorem: (Conlon Fox S.) For fixed n and large q, f (n, q) q q o(q) } n-1 times. f (n, 2) 2 2 } n-4 times. For q 5, q q! f (3, q) 5 2 q q!.
41 Tight lower bound for large alphabets Theorem: (Conlon Fox S.) For fixed n and large q, f (n, q) q q o(q) } n-1 times.
42 Tight lower bound for large alphabets Theorem: (Conlon Fox S.) For fixed n and large q, f (n, q) q q o(q) } n-1 times. Ingredients of the proof: Analyze a certain random construction using Lovász Local Lemma to show that there are q qq o(q) words avoiding Z 3.
43 Tight lower bound for large alphabets Theorem: (Conlon Fox S.) For fixed n and large q, f (n, q) q q o(q) } n-1 times. Ingredients of the proof: Analyze a certain random construction using Lovász Local Lemma to show that there are q qq o(q) words avoiding Z 3. To go from n to n + 1 develop an iterative step-up construction which gains an extra exponential at every step.
44 The step-up construction Lemma: Let S(n, q) denote the set of all words w over an alphabet of size q which avoid Z n and have a distinguished letter, say d, such that any subword of w not containing the letter d avoids Z n 1. Then S(n + 1, q + 2) S(n, q)!
45 The step-up construction Lemma: Let S(n, q) denote the set of all words w over an alphabet of size q which avoid Z n and have a distinguished letter, say d, such that any subword of w not containing the letter d avoids Z n 1. Then S(n + 1, q + 2) S(n, q)! Proof: Let m = S(n, q) and let w 1, w 2,..., w m be one of the m! orderings of the words in S(n, q).
46 The step-up construction Lemma: Let S(n, q) denote the set of all words w over an alphabet of size q which avoid Z n and have a distinguished letter, say d, such that any subword of w not containing the letter d avoids Z n 1. Then S(n + 1, q + 2) S(n, q)! Proof: Let m = S(n, q) and let w 1, w 2,..., w m be one of the m! orderings of the words in S(n, q). If c is the distinguished letter in w i for each i, let u i be the word formed from w i by replacing c with c 1 if i is odd and c 0 if i is even.
47 The step-up construction Lemma: Let S(n, q) denote the set of all words w over an alphabet of size q which avoid Z n and have a distinguished letter, say d, such that any subword of w not containing the letter d avoids Z n 1. Then S(n + 1, q + 2) S(n, q)! Proof: Let m = S(n, q) and let w 1, w 2,..., w m be one of the m! orderings of the words in S(n, q). If c is the distinguished letter in w i for each i, let u i be the word formed from w i by replacing c with c 1 if i is odd and c 0 if i is even. Then consider the word where d is a new letter. u 1 du 2 d... du m
48 The step-up construction Lemma: Let S(n, q) denote the set of all words w over an alphabet of size q which avoid Z n and have a distinguished letter, say d, such that any subword of w not containing the letter d avoids Z n 1. Then S(n + 1, q + 2) S(n, q)! Proof: Let m = S(n, q) and let w 1, w 2,..., w m be one of the m! orderings of the words in S(n, q). If c is the distinguished letter in w i for each i, let u i be the word formed from w i by replacing c with c 1 if i is odd and c 0 if i is even. Then consider the word u 1 du 2 d... du m where d is a new letter. This word contains q + 2 letters and we claim that it is in S(n + 1, q + 2).
49 An open problem Examples: P = xxx is 2-avoidable but not 1-avoidable.
50 An open problem Examples: P = xxx is 2-avoidable but not 1-avoidable. P = xx is 3-avoidable but not 2-avoidable.
51 An open problem Examples: P = xxx is 2-avoidable but not 1-avoidable. P = xx is 3-avoidable but not 2-avoidable. P = xyayzbzxcyxdxz is 4-avoidable but not 3-avoidable. (Baker McNulty Taylor 1989)
52 An open problem Examples: P = xxx is 2-avoidable but not 1-avoidable. P = xx is 3-avoidable but not 2-avoidable. P = xyayzbzxcyxdxz is 4-avoidable but not 3-avoidable. (Baker McNulty Taylor 1989) P = xyaxzbyxcyzdzwxewzw is 5-avoidable but not 4-avoidable. (Clark 2004)
53 An open problem Examples: P = xxx is 2-avoidable but not 1-avoidable. P = xx is 3-avoidable but not 2-avoidable. P = xyayzbzxcyxdxz is 4-avoidable but not 3-avoidable. (Baker McNulty Taylor 1989) P = xyaxzbyxcyzdzwxewzw is 5-avoidable but not 4-avoidable. (Clark 2004) Problem: Do there exist patterns P which are q-unavoidable but (q + 1)-avoidable for every integer q 1?
54 An open problem Examples: P = xxx is 2-avoidable but not 1-avoidable. P = xx is 3-avoidable but not 2-avoidable. P = xyayzbzxcyxdxz is 4-avoidable but not 3-avoidable. (Baker McNulty Taylor 1989) P = xyaxzbyxcyzdzwxewzw is 5-avoidable but not 4-avoidable. (Clark 2004) Problem: Do there exist patterns P which are q-unavoidable but (q + 1)-avoidable for every integer q 1? Remark: No such words have been found so far for q 5.
55
Tower-type bounds for unavoidable patterns in words
Tower-type bounds for unavoidable patterns in words David Conlon Jacob Fox Benny Sudakov Abstract A word w is said to contain the pattern P if there is a way to substitute a nonempty word for each letter
BOUNDS ON ZIMIN WORD AVOIDANCE
BOUNDS ON ZIMIN WORD AVOIDANCE JOSHUA COOPER* AND DANNY RORABAUGH* Abstract. How long can a word be that avoids the unavoidable? Word W encounters word V provided there is a homomorphism φ defined by mapping
The Erdős-Hajnal hypergraph Ramsey problem
The Erdős-Hajnal hypergraph Ramsey problem Dhruv Mubayi Andrew Suk February 28, 2016 Abstract Given integers 2 t k +1 n, let g k (t, n) be the minimum N such that every red/blue coloring of the k-subsets
Off-diagonal hypergraph Ramsey numbers
Off-diagonal hypergraph Ramsey numbers Dhruv Mubayi Andrew Suk Abstract The Ramsey number r k (s, n) is the minimum such that every red-blue coloring of the k- subsets of {1,..., } contains a red set of
Paul Erdős and Graph Ramsey Theory
Paul Erdős and Graph Ramsey Theory Benny Sudakov ETH and UCLA Ramsey theorem Ramsey theorem Definition: The Ramsey number r(s, n) is the minimum N such that every red-blue coloring of the edges of a complete
Constructions in Ramsey theory
Constructions in Ramsey theory Dhruv Mubayi Andrew Suk Abstract We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4-uniform
Variants of the Erdős-Szekeres and Erdős-Hajnal Ramsey problems
Variants of the Erdős-Szekeres and Erdős-Hajnal Ramsey problems Dhruv Mubayi December 19, 2016 Abstract Given integers l, n, the lth power of the path P n is the ordered graph Pn l with vertex set v 1
Hypergraph Ramsey numbers
Hypergraph Ramsey numbers David Conlon Jacob Fox Benny Sudakov Abstract The Ramsey number r k (s, n is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains a red
New lower bounds for hypergraph Ramsey numbers
New lower bounds for hypergraph Ramsey numbers Dhruv Mubayi Andrew Suk Abstract The Ramsey number r k (s, n) is the minimum N such that for every red-blue coloring of the k-tuples of {1,..., N}, there
Ramsey-type results for semi-algebraic relations
Ramsey-type results for semi-algebraic relations David Conlon Jacob Fox János Pach Benny Sudakov Andrew Suk Abstract A k-ary semi-algebraic relation E on R d is a subset of R kd, the set of k-tuples of
Toward the Combinatorial Limit Theory of free Words
University of South Carolina Scholar Commons Theses and Dissertations 015 Toward the Combinatorial Limit Theory of free Words Danny Rorabaugh University of South Carolina Follow this and additional works
PanHomc'r I'rui;* :".>r '.a'' W"»' I'fltolt. 'j'l :. r... Jnfii<on. Kslaiaaac. <.T i.. %.. 1 >
5 28 (x / &» )»(»»» Q ( 3 Q» (» ( (3 5» ( q 2 5 q 2 5 5 8) 5 2 2 ) ~ ( / x {» /»»»»» (»»» ( 3 ) / & Q ) X ] Q & X X X x» 8 ( &» 2 & % X ) 8 x & X ( #»»q 3 ( ) & X 3 / Q X»»» %» ( z 22 (»» 2» }» / & 2 X
PhD Seminar on Discrete and Applicable Mathematics in 2017
PhD Seminar on Discrete and Applicable Mathematics in 2017 Seminars are listed in reverse chronological order, most recent first. Friday 8 December - Benny Sudakov (ETH) Unavoidable patterns in words A
arxiv: v1 [cs.dm] 14 Oct 2016
![arxiv: v1 [cs.dm] 14 Oct 2016](/thumbs/74/70202760.jpg "arxiv: v1 [cs.dm] 14 Oct 2016")
Avoidability of circular formulas Guilhem Gamard a, Pascal Ochem a,b, Gwenaël Richomme a,c, Patrice Séébold a,c arxiv:1610.039v1 [cs.dm] 1 Oct 2016 Abstract a LIRMM, Université de Montpellier and CNRS,
Highly nonrepetitive sequences: winning strategies from the Local Lemma
Highly nonrepetitive sequences: winning strategies from the Local Lemma Wesley Pegden October 24, 2009 Abstract We prove game-theoretic versions of several classical results on nonrepetitive sequences,
Unary Pattern Avoidance in Partial Words Dense with Holes
Unary Pattern Avoidance in Partial Words Dense with Holes F Blanchet-Sadri 1 Kevin Black 2 Andrew Zemke 3 1 University of North Carolina at Greensboro 2 Harvey Mudd College 3 Rochester Institute of Technology
Marcin Witkowski. Nonrepetitive sequences on arithmetic progressions. Uniwersytet A. Mickiewicza w Poznaniu
Marcin Witkowski Uniwersytet A. Mickiewicza w Poznaniu Nonrepetitive sequences on arithmetic progressions Praca semestralna nr 2 (semestr zimowy 2010/11) Opiekun pracy: Jarosław Grytczuk NONREPETITIVE
Doubled patterns are 3-avoidable
Doubled patterns are 3-avoidable arxiv:1510.01753v1 [cs.dm] 6 Oct 2015 Pascal Ochem LIRMM, Université de Montpellier, CNRS Montpellier, France ochem@lirmm.fr August 31, 2018 Abstract In combinatorics on
Ordered Ramsey numbers
Ordered Ramsey numbers David Conlon Jacob Fox Choongbum Lee Benny Sudakov Abstract Given a labeled graph H with vertex set {1, 2,..., n}, the ordered Ramsey number r < (H) is the minimum N such that every
Erdős-Szekeres-type theorems for monotone paths and convex bodies
Erdős-Szekeres-type theorems for monotone paths and convex bodies Jacob Fox János Pach Benny Sudakov Andrew Suk Dedicated to the 75th anniversary of the publication of the Happy Ending Theorem Abstract
Cycle lengths in sparse graphs
Cycle lengths in sparse graphs Benny Sudakov Jacques Verstraëte Abstract Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value
Avoidability of formulas with two variables
Avoidability of formulas with two variables arxiv:1606.03955v2 [cs.dm] 13 Oct 2016 Pascal Ochem and Matthieu Rosenfeld October 14, 2016 Abstract In combinatorics on words, a word w over an alphabet Σ is
Ramsey-type problem for an almost monochromatic K 4
Ramsey-type problem for an almost monochromatic K 4 Jacob Fox Benny Sudakov Abstract In this short note we prove that there is a constant c such that every k-edge-coloring of the complete graph K n with
Avoiding Approximate Squares
Avoiding Approximate Squares Narad Rampersad School of Computer Science University of Waterloo 13 June 2007 (Joint work with Dalia Krieger, Pascal Ochem, and Jeffrey Shallit) Narad Rampersad (University
On avoidability of formulas with reversal
arxiv:1703.10522v1 [math.co] 30 Mar 2017 On avoidability of formulas with reversal James Currie, Lucas Mol, and Narad ampersad Abstract While a characterization of unavoidable formulas (without reversal)
On representable graphs
On representable graphs Sergey Kitaev and Artem Pyatkin 3rd November 2005 Abstract A graph G = (V, E) is representable if there exists a word W over the alphabet V such that letters x and y alternate in
arxiv: v1 [math.co] 2 Dec 2013
![arxiv: v1 [math.co] 2 Dec 2013](/thumbs/94/119265611.jpg "arxiv: v1 [math.co] 2 Dec 2013")
What is Ramsey-equivalent to a clique? Jacob Fox Andrey Grinshpun Anita Liebenau Yury Person Tibor Szabó arxiv:1312.0299v1 [math.co] 2 Dec 2013 November 4, 2018 Abstract A graph G is Ramsey for H if every
Avoidability of formulas with two variables
Avoidability of formulas with two variables Pascal Ochem and Matthieu Rosenfeld Submitted: October, 2017; Accepted: XX; Published: XX Mathematics Subject Classifications: 68R15 Abstract In combinatorics
On the grid Ramsey problem and related questions
On the grid Ramsey problem and related questions David Conlon Jacob Fox Choongbum Lee Benny Sudakov Abstract The Hales Jewett theorem is one of the pillars of Ramsey theory, from which many other results
Complementary Ramsey numbers, graph factorizations and Ramsey graphs
Complementary Ramsey numbers, graph factorizations and Ramsey graphs Akihiro Munemasa Tohoku University joint work with Masashi Shinohara May 30, 2017, Tohoku University 1st Tohoku-Bandung Bilateral Workshop:
Ramsey Theory. May 24, 2015
Ramsey Theory May 24, 2015 1 König s Lemma König s Lemma is a basic tool to move between finite and infinite combinatorics. To be concise, we use the notation [k] = {1, 2,..., k}, and [X] r will denote
Avoidable Formulas in Combinatorics on Words
University of California Los Angeles Avoidable Formulas in Combinatorics on Words A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics
Ramsey theory. Andrés Eduardo Caicedo. Graduate Student Seminar, October 19, Department of Mathematics Boise State University
Andrés Eduardo Department of Mathematics Boise State University Graduate Student Seminar, October 19, 2011 Thanks to the NSF for partial support through grant DMS-0801189. My work is mostly in set theory,
Three Proofs of the Hypergraph Ramsey Theorem (An. Exposition)
Three Proofs of the Hypergraph Ramsey Theorem (An Exposition William Gasarch Univ. of MD at College Park Andy Parrish Univ. of CA at San Diego Sandow Sinai Poolesville High School Abstract Ramsey, Erdős-Rado,
Induced Ramsey-type theorems
Induced Ramsey-type theorems Jacob Fox Benny Sudakov Abstract We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and
The Rainbow Turán Problem for Even Cycles
The Rainbow Turán Problem for Even Cycles Shagnik Das University of California, Los Angeles Aug 20, 2012 Joint work with Choongbum Lee and Benny Sudakov Plan 1 Historical Background Turán Problems Colouring
Some Problems in Graph Ramsey Theory. Andrey Vadim Grinshpun
Some Problems in Graph Ramsey Theory by Andrey Vadim Grinshpun Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics
A note on Gallai-Ramsey number of even wheels
A note on Gallai-Ramsey number of even wheels Zi-Xia Song a, Bing Wei b, Fangfang Zhang c,a, and Qinghong Zhao b a Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA b Department
Saturation numbers for Ramsey-minimal graphs
Saturation numbers for Ramsey-minimal graphs Martin Rolek and Zi-Xia Song Department of Mathematics University of Central Florida Orlando, FL 3816 August 17, 017 Abstract Given graphs H 1,..., H t, a graph
RMT 2014 Power Round Solutions February 15, 2014
Introduction This Power Round develops the many and varied properties of the Thue-Morse sequence, an infinite sequence of 0s and 1s which starts 0, 1, 1, 0, 1, 0, 0, 1,... and appears in a remarkable number
Vertex colorings of graphs without short odd cycles
Vertex colorings of graphs without short odd cycles Andrzej Dudek and Reshma Ramadurai Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 1513, USA {adudek,rramadur}@andrew.cmu.edu
De Bruijn sequences on primitive words and squares
De Bruijn sequences on primitive words and squares arxiv:0904.3997v2 [math.co] 24 Nov 2009 Yu-Hin Au Department of Combinatorics & Optimization yau@uwaterloo.ca November 24, 2009 Abstract We present a
RAMSEY THEORY. 1 Ramsey Numbers
RAMSEY THEORY 1 Ramsey Numbers Party Problem: Find the minimum number R(k, l) of guests that must be invited so that at least k will know each other or at least l will not know each other (we assume that
arxiv: v2 [math.co] 20 Jun 2018
![arxiv: v2 [math.co] 20 Jun 2018](/thumbs/91/107590130.jpg "arxiv: v2 [math.co] 20 Jun 2018")
ON ORDERED RAMSEY NUMBERS OF BOUNDED-DEGREE GRAPHS MARTIN BALKO, VÍT JELÍNEK, AND PAVEL VALTR arxiv:1606.0568v [math.co] 0 Jun 018 Abstract. An ordered graph is a pair G = G, ) where G is a graph and is
Combinatorics on Words:
Combinatorics on Words: Applications to Number Theory and Ramsey Theory Narad Rampersad Department of Mathematics and Statistics University of Winnipeg 9 May 2008 Narad Rampersad (University of Winnipeg)
arxiv: v1 [math.co] 27 Aug 2008
![arxiv: v1 [math.co] 27 Aug 2008](/thumbs/93/112338032.jpg "arxiv: v1 [math.co] 27 Aug 2008")
Hypergraph Ramsey numbers David Conlon Jacob Fox Benny Sudakov arxiv:0808.3760v1 [math.co] 27 Aug 2008 Abstract The Ramsey number r k (s,n is the minimum N such that everyred-bluecoloringofthe k-tuples
Explicit Construction of Small Folkman Graphs
Michael Spectral Graph Theory Final Presentation April 17, 2017 Notation Rado s Arrow Consider two graphs G and H. Then G (H) p is the statement that if the edges of G are p-colored, then there exists
Pattern-Matching for Strings with Short Descriptions
Pattern-Matching for Strings with Short Descriptions Marek Karpinski marek@cs.uni-bonn.de Department of Computer Science, University of Bonn, 164 Römerstraße, 53117 Bonn, Germany Wojciech Rytter rytter@mimuw.edu.pl
Binary words containing infinitely many overlaps
Binary words containing infinitely many overlaps arxiv:math/0511425v1 [math.co] 16 Nov 2005 James Currie Department of Mathematics University of Winnipeg Winnipeg, Manitoba R3B 2E9 (Canada) j.currie@uwinnipeg.ca
The subword complexity of a class of infinite binary words
arxiv:math/0512256v1 [math.co] 13 Dec 2005 The subword complexity of a class of infinite binary words Irina Gheorghiciuc November 16, 2018 Abstract Let A q be a q-letter alphabet and w be a right infinite
A ternary square-free sequence avoiding factors equivalent to abcacba
A ternary square-free sequence avoiding factors equivalent to abcacba James Currie Department of Mathematics & Statistics University of Winnipeg Winnipeg, MB Canada R3B 2E9 j.currie@uwinnipeg.ca Submitted:
Avoidability of Formulas with Two Variables
Avoidability of Formulas with Two Variables Pascal Ochem, Matthieu Rosenfeld To cite this version: Pascal Ochem, Matthieu Rosenfeld. Avoidability of Formulas with Two Variables. DLT: Developments in Language
Recursive Definitions
Recursive Definitions Example: Give a recursive definition of a n. a R and n N. Basis: n = 0, a 0 = 1. Recursion: a n+1 = a a n. Example: Give a recursive definition of n i=0 a i. Let S n = n i=0 a i,
A generalization of Thue freeness for partial words. By: Francine Blanchet-Sadri, Robert Mercaş, and Geoffrey Scott
A generalization of Thue freeness for partial words By: Francine Blanchet-Sadri, Robert Mercaş, and Geoffrey Scott F. Blanchet-Sadri, R. Mercas and G. Scott, A Generalization of Thue Freeness for Partial
A survey of hypergraph Ramsey problems
A survey of hypergraph Ramsey problems Dhruv Mubayi Andrew Suk Abstract The classical hypergraph Ramsey number r k (s, n) is the minimum N such that for every redblue coloring of the k-tuples of {1,...,
Patterns in Words Related to DNA Rearrangements
University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School June 2017 Patterns in Words Related to DNA Rearrangements Lukas Nabergall University of South Florida, lnabergall@mail.usf.edu
ON PATTERNS OCCURRING IN BINARY ALGEBRAIC NUMBERS
ON PATTERNS OCCURRING IN BINARY ALGEBRAIC NUMBERS B. ADAMCZEWSKI AND N. RAMPERSAD Abstract. We prove that every algebraic number contains infinitely many occurrences of 7/3-powers in its binary expansion.
About Duval Extensions
About Duval Extensions Tero Harju Dirk Nowotka Turku Centre for Computer Science, TUCS Department of Mathematics, University of Turku June 2003 Abstract A word v = wu is a (nontrivial) Duval extension
Some Variations on a Theme of Irina Mel nichuk Concerning the Avoidability of Patterns in Strings of Symbols
Some Variations on a Theme of Irina Mel nichuk Concerning the Avoidability of Patterns in Strings of Symbols George F. McNulty Department of Mathematics University of South Carolina Columbia, SC 908, U.S.A.
A conjecture on the alphabet size needed to produce all correlation classes of pairs of words
A conjecture on the alphabet size needed to produce all correlation classes of pairs of words Paul Leopardi Thanks: Jörg Arndt, Michael Barnsley, Richard Brent, Sylvain Forêt, Judy-anne Osborn. Mathematical
Monochromatic Boxes in Colored Grids
Monochromatic Boxes in Colored Grids Joshua Cooper, Stephen Fenner, and Semmy Purewal October 16, 008 Abstract A d-dimensional grid is a set of the form R = [a 1] [a d ] A d- dimensional box is a set of
Theorem (Special Case of Ramsey s Theorem) R(k, l) is finite. Furthermore, it satisfies,
Math 16A Notes, Wee 6 Scribe: Jesse Benavides Disclaimer: These notes are not nearly as polished (and quite possibly not nearly as correct) as a published paper. Please use them at your own ris. 1. Ramsey
PGSS Discrete Math Solutions to Problem Set #4. Note: signifies the end of a problem, and signifies the end of a proof.
PGSS Discrete Math Solutions to Problem Set #4 Note: signifies the end of a problem, and signifies the end of a proof. 1. Prove that for any k N, there are k consecutive composite numbers. (Hint: (k +
Packing nearly optimal Ramsey R(3, t) graphs
Packing nearly optimal Ramsey R(3, t) graphs He Guo Joint work with Lutz Warnke Context of this talk Ramsey number R(s, t) R(s, t) := minimum n N such that every red/blue edge-coloring of complete n-vertex
Model-theoretic distality and incidence combinatorics
Model-theoretic distality and incidence combinatorics Artem Chernikov UCLA Model Theory and Combinatorics workshop IHP, Paris, France Jan 29, 2018 Intro I will discuss generalizations of various results
Induced subgraphs with many repeated degrees
Induced subgraphs with many repeated degrees Yair Caro Raphael Yuster arxiv:1811.071v1 [math.co] 17 Nov 018 Abstract Erdős, Fajtlowicz and Staton asked for the least integer f(k such that every graph with
Square-free Strings Over Alphabet Lists
Square-free Strings Over Alphabet Lists [Updated: 08/10/015 11:31 - v.3] Neerja Mhaskar 1 and Michael Soltys 1 McMaster University Dept. of Computing & Software 180 Main Street West Hamilton, Ontario L8S
Word-representability of line graphs
Word-representability of line graphs Sergey Kitaev,1,3, Pavel Salimov,1,2, Christopher Severs,1, and Henning Úlfarsson,1 1 Reykjavik University, School of Computer Science, Menntavegi 1, 101 Reykjavik,
The Effect of Inequalities on Partition Regularity of Linear Homogenous Equations
The Effect of Inequalities on Partition Regularity of Linear Homogenous Equations Kavish Gandhi and Noah Golowich Mentor: Laszlo Miklos Lovasz MIT-PRIMES May 18, 2013 1 / 20 Kavish Gandhi and Noah Golowich
Packing nearly optimal Ramsey R(3, t) graphs
Packing nearly optimal Ramsey R(3, t) graphs He Guo Georgia Institute of Technology Joint work with Lutz Warnke Context of this talk Ramsey number R(s, t) R(s, t) := minimum n N such that every red/blue
Exact Bounds for Some Hypergraph Saturation Problems
Exact Bounds for Some Hypergraph Saturation Problems Guy Moshkovitz Asaf Shapira Abstract Let W n (p, q denote the minimum number of edges in an n n bipartite graph G on vertex sets X, Y that satisfies
Non-repetitive Tilings
Non-repetitive Tilings James D. Currie Department of Mathematics and Statistics University of Winnipeg Winnipeg, Manitoba Canada R3B 2E9 Fax: (204)-786-1824 E-mail: currie@uwpg02.uwinnipeg.ca Jamie Simpson
HW6 Solutions. Micha l Dereziński. March 20, 2015
HW6 Solutions Micha l Dereziński March 20, 2015 1 Exercise 5.5 (a) The PDA accepts odd-length strings whose middle symbol is a and whose other letters are as and bs. Its diagram is below. b, Z 0 /XZ 0
A sequence of triangle-free pseudorandom graphs
A sequence of triangle-free pseudorandom graphs David Conlon Abstract A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge density significantly higher than one
Combinatorics on Words with Applications
Combinatorics on Words with Applications Mark V. Sapir December 11, 1993 Contents 1 Introduction 2 1.1 Main De nitions : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 About the Course : : : : :
RAINBOW 3-TERM ARITHMETIC PROGRESSIONS. Veselin Jungić Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada.
RAINBOW 3-TERM ARITHMETIC PROGRESSIONS Veselin Jungić Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada vjungic@sfu.ca Radoš Radoičić Department of Mathematics, MIT, Cambridge,
An Algorithmic Proof of the Lopsided Lovász Local Lemma (simplified and condensed into lecture notes)
An Algorithmic Proof of the Lopsided Lovász Local Lemma (simplified and condensed into lecture notes) Nicholas J. A. Harvey University of British Columbia Vancouver, Canada nickhar@cs.ubc.ca Jan Vondrák
The Lopsided Lovász Local Lemma
Joint work with Linyuan Lu and László Székely Georgia Southern University April 27, 2013 The lopsided Lovász local lemma can establish the existence of objects satisfying several weakly correlated conditions
On rainbow arithmetic progressions
On rainbow arithmetic progressions Maria Axenovich Department of Mathematics Iowa State University USA axenovic@math.iastate.edu Dmitri Fon-Der-Flaass Department of Mathematics University of Illinois at
Probabilistic Method. Benny Sudakov. Princeton University
Probabilistic Method Benny Sudakov Princeton University Rough outline The basic Probabilistic method can be described as follows: In order to prove the existence of a combinatorial structure with certain
Binomial Coefficient Identities/Complements
Binomial Coefficient Identities/Complements CSE21 Fall 2017, Day 4 Oct 6, 2017 https://sites.google.com/a/eng.ucsd.edu/cse21-fall-2017-miles-jones/ permutation P(n,r) = n(n-1) (n-2) (n-r+1) = Terminology
Probabilistic Methods in Combinatorics Lecture 6
Probabilistic Methods in Combinatorics Lecture 6 Linyuan Lu University of South Carolina Mathematical Sciences Center at Tsinghua University November 16, 2011 December 30, 2011 Balance graphs H has v vertices
Two-coloring random hypergraphs
Two-coloring random hypergraphs Dimitris Achlioptas Jeong Han Kim Michael Krivelevich Prasad Tetali December 17, 1999 Technical Report MSR-TR-99-99 Microsoft Research Microsoft Corporation One Microsoft
Abelian Pattern Avoidance in Partial Words
Abelian Pattern Avoidance in Partial Words F. Blanchet-Sadri 1 Benjamin De Winkle 2 Sean Simmons 3 July 22, 2013 Abstract Pattern avoidance is an important topic in combinatorics on words which dates back
On the size-ramsey numbers for hypergraphs. A. Dudek, S. La Fleur and D. Mubayi
On the size-ramsey numbers for hypergraphs A. Dudek, S. La Fleur and D. Mubayi REPORT No. 48, 013/014, spring ISSN 1103-467X ISRN IML-R- -48-13/14- -SE+spring On the size-ramsey number of hypergraphs Andrzej
Monochromatic Solutions to Equations with Unit Fractions
Monochromatic Solutions to Equations with Unit Fractions Tom C. Brown and Voijtech Rödl Citation data T.C. Brown and V. Rödl, Monochromatic solutions to equations with unit fractions, Bull. Aus. Math.
arxiv: v1 [cs.dm] 13 Feb 2010
![arxiv: v1 [cs.dm] 13 Feb 2010](/thumbs/76/73124380.jpg "arxiv: v1 [cs.dm] 13 Feb 2010")
Properties of palindromes in finite words arxiv:1002.2723v1 [cs.dm] 13 Feb 2010 Mira-Cristiana ANISIU Valeriu ANISIU Zoltán KÁSA Abstract We present a method which displays all palindromes of a given length
Chapter 5: Integer Compositions and Partitions and Set Partitions
Chapter 5: Integer Compositions and Partitions and Set Partitions Prof. Tesler Math 184A Winter 2017 Prof. Tesler Ch. 5: Compositions and Partitions Math 184A / Winter 2017 1 / 32 5.1. Compositions A strict
Notes on Continued Fractions for Math 4400
. Continued fractions. Notes on Continued Fractions for Math 4400 The continued fraction expansion converts a positive real number α into a sequence of natural numbers. Conversely, a sequence of natural
Pigeonhole Principle and Ramsey Theory
Pigeonhole Principle and Ramsey Theory The Pigeonhole Principle (PP) has often been termed as one of the most fundamental principles in combinatorics. The familiar statement is that if we have n pigeonholes
Open Problems in Automata Theory: An Idiosyncratic View
Open Problems in Automata Theory: An Idiosyncratic View Jeffrey Shallit School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1 Canada shallit@cs.uwaterloo.ca http://www.cs.uwaterloo.ca/~shallit
Please give details of your answer. A direct answer without explanation is not counted.
Please give details of your answer. A direct answer without explanation is not counted. Your answers must be in English. Please carefully read problem statements. During the exam you are not allowed to
Every binary word is, almost, a shuffle of twin subsequences a theorem of Axenovich, Person and Puzynina
Every binary word is, almost, a shuffle of twin subsequences a theorem of Axenovich, Person and Puzynina Martin Klazar August 17, 2015 A twin in a word u = a 1 a 2... a n is a pair (u 1, u 2 ) of disjoint
Recurrence Relations and Recursion: MATH 180
Recurrence Relations and Recursion: MATH 180 1: Recursively Defined Sequences Example 1: The sequence a 1,a 2,a 3,... can be defined recursively as follows: (1) For all integers k 2, a k = a k 1 + 1 (2)
Ramsey theory. Andrés Eduardo Caicedo. Undergraduate Math Seminar, March 22, Department of Mathematics Boise State University
Andrés Eduardo Department of Mathematics Boise State University Undergraduate Math Seminar, March 22, 2012 Thanks to the NSF for partial support through grant DMS-0801189. My work is mostly in set theory,
Every Monotone Graph Property is Testable
Every Monotone Graph Property is Testable Noga Alon Asaf Shapira Abstract A graph property is called monotone if it is closed under removal of edges and vertices. Many monotone graph properties are some
LOWELL WEEKI.Y JOURINAL
/ $ 8) 2 {!»!» X ( (!!!?! () ~ x 8» x /»!! $?» 8! ) ( ) 8 X x /! / x 9 ( 2 2! z»!!»! ) / x»! ( (»»!» [ ~!! 8 X / Q X x» ( (!»! Q ) X x X!! (? ( ()» 9 X»/ Q ( (X )!» / )! X» x / 6!»! }? ( q ( ) / X! 8 x»
Ramsey theory and the geometry of Banach spaces
Ramsey theory and the geometry of Banach spaces Pandelis Dodos University of Athens Maresias (São Paulo), August 25 29, 2014 1.a. The Hales Jewett theorem The following result is due to Hales & Jewett
Induced Graph Ramsey Theory
Induced Graph Ramsey Theory Marcus Schaefer School of CTI DePaul University 243 South Wabash Avenue Chicago, Illinois 60604, USA schaefer@csdepauledu June 28, 2000 Pradyut Shah Department of Computer Science
Words generated by cellular automata
Words generated by cellular automata Eric Rowland University of Waterloo (soon to be LaCIM) November 25, 2011 Eric Rowland (Waterloo) Words generated by cellular automata November 25, 2011 1 / 38 Outline