[Paper Review] On the maximum number of integer colourings with forbidden monochromatic sums

Hong Liu, Maryam Sharifzadeh+1 more|arXiv (Cornell University)|Sep 27, 2017

Limits and Structures in Graph Theory16 references3 citations

TL;DR

This paper investigates the maximum number of r-colourings of subsets of {1, ..., n} such that no colour class contains a Schur triple (i.e., a solution to x + y = z). For r = 2, it proves f(n,2) = 2^{⌈n/2⌉}, with extremal sets being the odd numbers or intervals of size ⌈n/2⌉. For r ≤ 5, it asymptotically determines log f(n,r) using linear programming, showing that extremal sets are close to the odd numbers or unions of intervals.

ABSTRACT

Let $f(n,r)$ denote the maximum number of colourings of $A \subseteq \lbrace 1,\ldots,n brace$ with $r$ colours such that each colour class is sum-free. Here, a sum is a subset $\lbrace x,y,z brace$ such that $x+y=z$. We show that $f(n,2) = 2^{\lceil n/2 ceil}$, and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of $f(n,r)$ for $r \leq 5$. Similar results were obtained by Hàn and Jiménez in the setting of finite abelian groups.

Motivation & Objective

• To determine f(n,r), the maximum number of r-colourings of subsets A ⊆ {1,…,n} such that each colour class is sum-free (contains no Schur triple).
• To extend the Erdős-Rothschild problem from graph theory to the sum-free setting, replacing cliques with Schur triples.
• To characterize extremal sets achieving f(n,r) and understand their structure for small r.
• To asymptotically determine log f(n,r) for r ≤ 5 using linear optimization techniques.
• To explore the stability of extremal sets and conjecture structural approximations for r = 4,5.

Proposed method

• Define f(n,r) as the maximum number of r-colourings of any subset A ⊆ {1,…,n} where each colour class is sum-free.
• Use extremal combinatorics and stability methods to analyze the structure of sets achieving maximum colourings.
• For r = 2, prove that the extremal sets are the odd numbers or intervals of size ⌈n/2⌉, yielding f(n,2) = 2^{⌈n/2⌉}.
• For r ≤ 5, apply linear programming to model constraints on density and sum-free properties, deriving asymptotic bounds on log f(n,r).
• Use a layering approach inspired by Pikhurko et al. to reduce the problem to finite optimization over sum-free configurations.
• Establish stability results by showing that extremal sets must be close (in symmetric difference) to the odd numbers or unions of intervals.

Experimental results

Research questions

• RQ1 What is the maximum number of 2-colourings of a subset of {1,…,n} such that no colour class contains a Schur triple?
• RQ2 For r ∈ {3,4,5}, what is the asymptotic growth of log f(n,r), and what structural properties do extremal sets have?
• RQ3 Can the extremal sets for f(n,r) be characterized as being close to the odd numbers or unions of intervals?
• RQ4 Is the trivial lower bound from sum-free sets tight for r ≤ 5, or do extremal graphs contain many Schur triples?
• RQ5 Can linear programming techniques be used to derive exact or asymptotic results for larger r, such as r = 6?

Key findings

• For r = 2, f(n,2) = 2^{⌈n/2⌉}, with extremal sets being the odd numbers or any interval of size ⌈n/2⌉.
• The extremal sets achieving f(n,2) are exactly the sum-free sets of maximum size, i.e., the odd numbers or intervals of size ⌈n/2⌉.
• For r ≤ 5, the logarithm of f(n,r) is asymptotically determined via linear programming, showing that log f(n,r) = h(r) · n + o(n) for some h(r).
• The extremal sets for r = 4 and r = 5 are structurally close to the odd numbers or the union of the odd numbers and a specific interval.
• Stability results suggest that any extremal set for r = 4 must be within o(n) symmetric difference of the odd numbers, I₂, or their union.
• For r ≥ 6, the method becomes significantly more complex due to the need to consider multiple types of sum-free configurations, leaving the problem open.