On Three Sets with Nondecreasing Diameter Carl Yerger Davidson - PowerPoint PPT Presentation

On Three Sets with Nondecreasing Diameter Carl Yerger Davidson College cayerger@davidson.edu Joint work with Daniel Bernstein, Davidson College and David Grynkiewicz, Karl-Franzens University of Graz May 13, 2011 Carl Yerger (Davidson College) On Three Sets... May 13, 2011 1 / 18

Background - The Pigeonhole Principle In 1961, Erd˝ os, Ginzburg and Ziv proved the following theorem that is the subject of many papers and generalizations. Theorem Let m ∈ N . Every sequence of 2 m − 1 elements from Z contains a subsequence of m elements whose sum is zero modulo m. Notice that this theorem is a generalization of the pigeonhole principle. For instance, if the sequence contains only the residues 0 and 1, then this theorem describes the situation of placing 2 m − 1 pigeons in 2 holes. Carl Yerger (Davidson College) On Three Sets... May 13, 2011 2 / 18

Preliminaries Some Definitions Carl Yerger (Davidson College) On Three Sets... May 13, 2011 3 / 18

Preliminaries Some Definitions Let ∆ : X − → C , where C is the set of colors. This could be integers, residues, etc. Carl Yerger (Davidson College) On Three Sets... May 13, 2011 3 / 18

Preliminaries Some Definitions Let ∆ : X − → C , where C is the set of colors. This could be integers, residues, etc. If C = { 1 , . . . , k } , ∆ is a k -coloring. Carl Yerger (Davidson College) On Three Sets... May 13, 2011 3 / 18

Preliminaries Some Definitions Let ∆ : X − → C , where C is the set of colors. This could be integers, residues, etc. If C = { 1 , . . . , k } , ∆ is a k -coloring. If set C = Z , we call ∆ a Z -coloring. Carl Yerger (Davidson College) On Three Sets... May 13, 2011 3 / 18

Preliminaries Some Definitions Let ∆ : X − → C , where C is the set of colors. This could be integers, residues, etc. If C = { 1 , . . . , k } , ∆ is a k -coloring. If set C = Z , we call ∆ a Z -coloring. A set X is called monochromatic if ∆( x ) = ∆( x ′ ) for all x , x ′ ∈ X . Carl Yerger (Davidson College) On Three Sets... May 13, 2011 3 / 18

Preliminaries Some Definitions Let ∆ : X − → C , where C is the set of colors. This could be integers, residues, etc. If C = { 1 , . . . , k } , ∆ is a k -coloring. If set C = Z , we call ∆ a Z -coloring. A set X is called monochromatic if ∆( x ) = ∆( x ′ ) for all x , x ′ ∈ X . In a Z - coloring of X , a subset Y of X is called zero-sum modulo n if � y ∈ Y ∆( y ) ≡ 0 mod n . Carl Yerger (Davidson College) On Three Sets... May 13, 2011 3 / 18

Preliminaries Some Definitions Let ∆ : X − → C , where C is the set of colors. This could be integers, residues, etc. If C = { 1 , . . . , k } , ∆ is a k -coloring. If set C = Z , we call ∆ a Z -coloring. A set X is called monochromatic if ∆( x ) = ∆( x ′ ) for all x , x ′ ∈ X . In a Z - coloring of X , a subset Y of X is called zero-sum modulo n if � y ∈ Y ∆( y ) ≡ 0 mod n . If a , b ∈ Z , then [ a , b ] is the set of integers { n ∈ Z | a ≤ n ≤ b } . Carl Yerger (Davidson College) On Three Sets... May 13, 2011 3 / 18

Preliminaries Some Definitions Let ∆ : X − → C , where C is the set of colors. This could be integers, residues, etc. If C = { 1 , . . . , k } , ∆ is a k -coloring. If set C = Z , we call ∆ a Z -coloring. A set X is called monochromatic if ∆( x ) = ∆( x ′ ) for all x , x ′ ∈ X . In a Z - coloring of X , a subset Y of X is called zero-sum modulo n if � y ∈ Y ∆( y ) ≡ 0 mod n . If a , b ∈ Z , then [ a , b ] is the set of integers { n ∈ Z | a ≤ n ≤ b } . For finite X ⊆ N , define the diameter of X , denoted by diam ( X ), to be diam ( X ) = max( X ) − min( X ). Carl Yerger (Davidson College) On Three Sets... May 13, 2011 3 / 18

A Coloring Problem Coloring Set-up Carl Yerger (Davidson College) On Three Sets... May 13, 2011 4 / 18

A Coloring Problem Coloring Set-up Let f ( s , r , k ) be the the smallest positive integer n such that for every coloring ∆ : [1 , n ] − → [1 , k ] there exist two subsets S 1 , S 2 of [1 , n ], which satisfy: Carl Yerger (Davidson College) On Three Sets... May 13, 2011 4 / 18

A Coloring Problem Coloring Set-up Let f ( s , r , k ) be the the smallest positive integer n such that for every coloring ∆ : [1 , n ] − → [1 , k ] there exist two subsets S 1 , S 2 of [1 , n ], which satisfy: ( a ) S 1 and S 2 are monochromatic ( b ) | S 1 | = s , | S 2 | = r , ( c ) max( S 1 ) < min( S 2 ), and ( d ) diam ( S 1 ) ≤ diam ( S 2 ). Theorem Carl Yerger (Davidson College) On Three Sets... May 13, 2011 4 / 18

A Coloring Problem Coloring Set-up Let f ( s , r , k ) be the the smallest positive integer n such that for every coloring ∆ : [1 , n ] − → [1 , k ] there exist two subsets S 1 , S 2 of [1 , n ], which satisfy: ( a ) S 1 and S 2 are monochromatic ( b ) | S 1 | = s , | S 2 | = r , ( c ) max( S 1 ) < min( S 2 ), and ( d ) diam ( S 1 ) ≤ diam ( S 2 ). Theorem In a paper of Bialostocki, Erd˝ os and Lefmann (1995), it was shown that f ( m , m , 2) = 5 m − 3 and f ( m , m , 3) = 9 m − 7. Carl Yerger (Davidson College) On Three Sets... May 13, 2011 4 / 18

A More General Coloring Problem Two Types of Problems (one in parentheses) Carl Yerger (Davidson College) On Three Sets... May 13, 2011 5 / 18

A More General Coloring Problem Two Types of Problems (one in parentheses) Let f ( s , r , Z )) ( f ( s , r , Z ) ∪ {∞} )) be the the smallest positive integer n such that for every coloring ∆ : [1 , n ] − → Z (∆ : [1 , n ] − → {∞} ∪ Z ), there exist two subsets S 1 , S 2 of [1 , n ], which satisfy: ( a ) S 1 is zero-sum mod s and S 2 is zero-sum mod r ( S 1 is either ∞ -monochromatic or zero-sum mod s and S 2 is either ∞ -monochromatic or zero-sum mod r ), ( b ) | S 1 | = s , | S 2 | = r , ( c ) max( S 1 ) < min( S 2 ), and ( d ) diam ( S 1 ) ≤ diam ( S 2 ). Theorem Carl Yerger (Davidson College) On Three Sets... May 13, 2011 5 / 18

A More General Coloring Problem Two Types of Problems (one in parentheses) Let f ( s , r , Z )) ( f ( s , r , Z ) ∪ {∞} )) be the the smallest positive integer n such that for every coloring ∆ : [1 , n ] − → Z (∆ : [1 , n ] − → {∞} ∪ Z ), there exist two subsets S 1 , S 2 of [1 , n ], which satisfy: ( a ) S 1 is zero-sum mod s and S 2 is zero-sum mod r ( S 1 is either ∞ -monochromatic or zero-sum mod s and S 2 is either ∞ -monochromatic or zero-sum mod r ), ( b ) | S 1 | = s , | S 2 | = r , ( c ) max( S 1 ) < min( S 2 ), and ( d ) diam ( S 1 ) ≤ diam ( S 2 ). Theorem The paper of Bialostocki, Erd˝ os and Lefmann, also shows that f ( m , m , 2) = f ( m , m , Z ) = 5 m − 3 and f ( m , m , 3) = f ( m , m , {∞} ∪ Z ) = 9 m − 7. Such theorems are known as zero-sum generalizations in the sense of Erd˝ os-Ginzburg-Ziv . Carl Yerger (Davidson College) On Three Sets... May 13, 2011 5 / 18

Lower Bound Constructions Lower bounds for 2-colorings and 3-colorings Carl Yerger (Davidson College) On Three Sets... May 13, 2011 6 / 18

Lower Bound Constructions Lower bounds for 2-colorings and 3-colorings To show that f ( m , m , 2) > 5 m − 4 consider: 21 m − 1 2 m − 1 1 m − 1 2 2 m − 2 Carl Yerger (Davidson College) On Three Sets... May 13, 2011 6 / 18

Lower Bound Constructions Lower bounds for 2-colorings and 3-colorings To show that f ( m , m , 2) > 5 m − 4 consider: 21 m − 1 2 m − 1 1 m − 1 2 2 m − 2 To see that f ( m , m , 3) > 9 m − 8, consider: 31 m − 1 2 m − 1 3 m − 1 1 m − 1 2 m − 1 1 m − 1 2 m − 1 3 2 m − 2 Lower bounds for EGZ generalizations Carl Yerger (Davidson College) On Three Sets... May 13, 2011 6 / 18

Lower Bound Constructions Lower bounds for 2-colorings and 3-colorings To show that f ( m , m , 2) > 5 m − 4 consider: 21 m − 1 2 m − 1 1 m − 1 2 2 m − 2 To see that f ( m , m , 3) > 9 m − 8, consider: 31 m − 1 2 m − 1 3 m − 1 1 m − 1 2 m − 1 1 m − 1 2 m − 1 3 2 m − 2 Lower bounds for EGZ generalizations To show that f ( m , m , Z ) > 5 m − 4 consider: 10 m − 1 1 m − 1 0 m − 1 1 2 m − 2 Carl Yerger (Davidson College) On Three Sets... May 13, 2011 6 / 18

Lower Bound Constructions Lower bounds for 2-colorings and 3-colorings To show that f ( m , m , 2) > 5 m − 4 consider: 21 m − 1 2 m − 1 1 m − 1 2 2 m − 2 To see that f ( m , m , 3) > 9 m − 8, consider: 31 m − 1 2 m − 1 3 m − 1 1 m − 1 2 m − 1 1 m − 1 2 m − 1 3 2 m − 2 Lower bounds for EGZ generalizations To show that f ( m , m , Z ) > 5 m − 4 consider: 10 m − 1 1 m − 1 0 m − 1 1 2 m − 2 To see that f ( m , m , {∞} ∪ Z ) > 9 m − 8, consider: ∞ 0 m − 1 1 m − 1 ∞ m − 1 0 m − 1 1 m − 1 0 m − 1 1 m − 1 ∞ 2 m − 2 Carl Yerger (Davidson College) On Three Sets... May 13, 2011 6 / 18

Generalization of these Problems What if the two sets are different sizes? (CY, 2005) For r ≥ s ≥ 2, f ( s , r , 2) =  5 s − 3 if s = r  4 s + r − 3 if s < r ≤ 2 s − 2 2 s + 2 r − 2 if r > 2 s − 2  For r ≥ s ≥ 3, f ( s , r , Z ) =  5 s − 3 if s = r  s 4 s + max( r , s + ( r , s ) − 1) − 3 if s < r ≤ 2 s − 2 2 s + 2 r − 2 if r > 2 s − 2  Note: for integers, s , r , let ( s , r ) be the greatest common factor of s and r . So (8 , 12) = 4. Carl Yerger (Davidson College) On Three Sets... May 13, 2011 7 / 18