$L(4) = 16$ for the number $L(n)$ of Eliahou and Revuelta

This is the goal of the mission: the exact value of $L(4)$.

A set of natural numbers is sumfree if it contains no $x, y, z$ with $x + y = z$ ($x = y$ allowed). For a finite sequence $A = (a_1, \dots, a_N)$ of positive integers, let $\hat A = \{a_i + \dots + a_j : 1 \le i \le j \le N\}$ be its set of block sums, $\mu(A) = (a_1 + \dots + a_N)/N$ its average, and $\operatorname{sdeg}(\hat A)$ the least $n \ge 1$ such that $\hat A$ is covered by $n$ sumfree sets. For $n \ge 2$, Eliahou and Revuelta define $L(n)$ as the least positive integer $L$ such that every sequence $A$ of positive integers with $|A| = L$ and $\mu(A) \le n$ satisfies $\operatorname{sdeg}(\hat A) \ge n$. Then

Eliahou and Revuelta prove $S(n-1) + 1 \le L(n) \le R_{n-1}(3) - 1$ (Proposition 5.3), which gives $14 \le L(4) \le 16$ from the Schur number $S(3) = 13$ and the Ramsey number $R_3(3) = 17$. They conjecture $L(n) = S(n-1) + 1$ for all $n \ge 2$ (Conjecture 5.6), which at $n = 4$ states $L(4) = 14$. The value $16$ shows that Conjecture 5.6 does not hold at $n = 4$, and that at $n = 4$ the upper bound of Proposition 5.3 is attained.

Formalization Note $L(4)$ is erL 4 of the Basic bundle, with the ambient set $\mathbb{N}$ and the Schur degree in the extended naturals. The defining set is nonempty for $n = 4$, so erL 4 is its least element.