No cover of $\hat A$ by $q \le k$ sumfree sets when $|A| \ge N - 1$ and $k$ colours force a monochromatic triangle on $N$ vertices

This is the cover form of Theorem 4.1 of Eliahou and Revuelta, with the Ramsey number replaced by any $N$ that has the monochromatic-triangle property.

Let $k, N \in \mathbb{N}$ and suppose that $\mathrm{TR}(k, N)$ holds: every colouring, with at most $k$ colours, of the pairs $x < y$ of a finite set of at least $N$ natural numbers has a monochromatic triangle. Let $A$ be a finite sequence of natural numbers with $N \le |A| + 1$, and let $q \le k$. Then the set $\hat A$ of block sums of $A$ (the sums of the nonempty runs of consecutive entries) is not covered by $q$ sumfree sets:

Here a set is sumfree if it contains no $x, y, z$ with $x + y = z$, $x = y$ allowed.

Eliahou and Revuelta state Theorem 4.1 in this form in their introduction: if $\hat A$ is covered by $n$ sumfree sets, then $|A| \le R_n(3) - 2$. Combined with the pigeonhole bound $\mathrm{TR}(k, \rho(k))$, this lemma gives $\operatorname{sdeg}(\hat A) \ge k + 1$ whenever $|A| \ge \rho(k) - 1$.

Formalization Note The paper states Theorem 4.1 for sequences in an arbitrary abelian group; here the entries are natural numbers, and they are not required to be positive. The hypothesis $|A| \ge N - 1$ is written $N \le |A| + 1$. The case $q = 0$ is included; it states that $\hat A$ is nonempty.