1. Chromatic Number of Distance Graphs Generated by the Sets { 2, 3, x, y } Daphne Liu and Aileen Sutedja Department of Mathematics California State Univ., Los Angeles

2. Distance Graphs Eggleton, Erdős, Skilton [1985 – 1987] Fix D a set of positive integers, the distance graph G(Z, D) has: Vertices: All integers Z Edges: D = { 1, 3, 4 } 0 123 4 5678

3. Example D = {1, 3, 5, 7, 9}. Then 0 123 4 5678 χ (D) = 2

4. 3 – Element Sets Let D = { a, b, c }, a < b < c. [Eggleton, Erdős, Skilton 1985] [Chen, Huang, Chang 1997][Voigt, 1999] [Zhu, 2002]

5. Chromatic Number of Distance Graphs If D is a subset of prime numbers then χ (D) ≤ 4. If D contains only odd numbers then χ (D) ≤ 2. If D contains no multiples of k, then χ (D) ≤ k.

6. 4 – Element Prime Sets For a prime set D = {2, 3, p, q}, χ (D) = 3 or 4. Question: For a prime set D = {2, 3, p, q}. Which sets D have χ (D) = 4 ?

7. Complete solutions on 4-element prime sets Let D be a prime set, D={2, 3, p, q}. Then χ (D) = 4 if and only if p, q are twin primes, or (p, q) is one of the following: (11,19), (11,23), (11, 37), (11, 41), (17, 29), (23, 31), (23, 41), (29, 37). [Voigt and Walther, 1994] [Eggleton, Erdős, Skilton 1990]

8. General 4 – element Sets D = { 2, 3, x, x+s }, x > 3. Voigt and Walther proved that χ (D) = 3 if s ≥ 10 and x ≥ s 2 – 6s +3. Kemnitz and Kolberg determined the chromatic number for all s < 10. Our aim: Completely solve this problem.

9. Theorems IfD = {1, 2, 3, 4m} or D = {x, y, y-x, y+x}, x and y are odd. Then χ (D) = 4. [Kemnitz & Marangio] [L. & Zhu] Let |D| = 4. Then χ (D) ≤ 4 unless D is the above two types. [Barajas & Serra 2008]

10. Main Tools – Useful Results Chang, L., Zhu, 1999Zhu, 2001 Let D = {2, 3, x, y}. Then

11. Density of Sequences w/ Missing Differences Let D be a set of positive integers. Example, D = {1, 4, 5}. “density” of this M(D) is 1/3. A sequence with missing differences of D, denoted by M(D), is one such that the absolute difference of any two terms does not fall in D. For instance, M(D) = {3, 6, 9, 12, 15, …} μ (D) = maximum density of an M(D). => μ ({1, 4, 5}) = 1/3.

12. Theorem [Chang, L., Zhu, 1999] For any finite set of integers D, χ f (G) is the fractional chromatic number.

13. Parameter involved in the Lonely Runner Conjecture For any real x, let || x || denote the shortest distance from x to an integer. For instance, ||3.2|| = 0.2 and ||4.9|| = 0.1. Let D be a set of real numbers, let t be any real number: ||D t|| : = min { || d t ||: d є D}. κ (D) : = sup { || D t ||: t є R}.

14. Example D = { 1, 3, 4 } ||(1/3) D|| = min {1/3, 0, 1/3} = 0 ||(1/4) D|| = min {1/4, 1/4, 0} = 0 ||(1/7) D|| = min {1/7, 3/7, 3/7} = 1/7 ||(2/7) D|| = min {2/7, 1/7, 1/7} = 1/7 ||(3/7) D|| = min {3/7, 2/7, 2/7} = 2/7 κ (D) = 2/7

15. Useful Lemmas – Lower Bounds Theorem:Let 0 < t <1. If for every D-sequence S there exists some n ≥ 0 such that S[n]/(n+1) ≤ t then μ(D) ≤ t. [Haralambis 1977] Corollary:If μ(D) ≥ 1/3, then there exists a D-sequence S so that S[n]/(n+1) ≥ 1/3 for all n ≥ 0.

16. Upper Bounds For any D, κ (D) = m/n, where n is the sum of two elements in D.

17. Idea 0 1/3

18. Theorem [L. and Sutedja 2011] D = { 2, 3, x, x+10 }. D = { 2, 3, 6, y }. D = { 2, 3, 10, y } D = { 2, 3, 4, y }.

19. Alternative Definition of κ (D)

20. Theorem [L. and Sutedja 2011] Let D ={ 2, 3, x, y }. Then χ (D)=3 for: X = 9, y ≥ 36 X = 11, y ≥ 66 X = 15, y ≥ 60 X = 16, y ≥ 48 X ≥ 53, y ≥ x - 11 X = 12, 13, 14, or x ≥ 17, and y ≥ 2 x

21. Theorem (continue) Let D = { 2, 3, x, x+s}, x ≥ 7 and s ≥ 11. Then χ (D) = 3, except (x, x+s) is one of: (9, 23), (11, 23), (11, 27), (11, 28), (11, 32), (11, 37), (11, 41), (11, 46), (15, 41), (16, 37), (17, 29), (18, 31), (23, 36), (23, 41), (24, 37), (28, 41), for which χ (D) = 4.

22. Open Problems Let D be a prime set with |D| = 5. For what sets D, we have χ (D) = 3? Let D be a set with |D| = 4. For what sets D, we have χ (D) = 3? Exact values of κ (D) and μ (D) for D = { 2, 3, x, y }.

23. Lonely Runner Conjecture Suppose k runners running on a circular field of circumference r. Suppose each runner keeps a constant speed and all runners have different speeds. A runner is called “lonely” at some moment if he or she has (circular) distance at least r/k apart from all other runners. Conjecture (Wills, Goddyn): For each runner, there exists some time that he or she is lonely.

24. Wills Conjecture For any D, Wills, Diophantine approximation, in German, 1967. Cusick and Pomerance, 1984. (True for |D| ≤ 4.) Chen, J. Number Theory, a generalized conjecture. Bohman, Holzman and Kleitman 2001. |D| = 5. Barajas & Serra 2008. |D| = 6.