A sufficient condition for first order non-definability of arrowing problems

We here present a sufficient condition for general arrowing problems to be non definable in first order logic, based in well known tools of finite model theory e.g. Hanf’s Theorem and known concepts in finite combinatorics, like senders and determiners.

💡 Research Summary

The paper investigates the first‑order (FO) definability of a broad class of arrowing problems, which are Ramsey‑type questions asking whether, for two given graphs G and H, every edge‑coloring of a sufficiently large host graph forces a monochromatic copy of G to embed into a copy of H. While the computational complexity of such problems has been extensively studied, their logical complexity—specifically whether they can be expressed by FO sentences—remains less understood.

The authors build their main result on Hanf’s Theorem, a cornerstone of finite model theory. Hanf’s Theorem states that two finite structures that have identical distributions of local neighborhoods up to a fixed radius r cannot be distinguished by any FO sentence whose quantifier depth is bounded by r. Consequently, if one can construct pairs of graph instances that differ globally but share the same r‑type distribution, any FO formula will evaluate them identically, proving FO‑non‑definability.

To operationalize this insight, the paper introduces two combinatorial gadgets: senders and determiners. A sender is a small subgraph that forces a particular coloring pattern locally; a determiner is a configuration that propagates the effect of the sender throughout the larger host graph. By carefully embedding copies of a fixed sender S and a fixed determiner D into an infinite family F of increasingly large graphs, the authors ensure that for any fixed radius r, the local neighborhoods around any vertex in the modified graph G′ look the same as those in another graph from the family, regardless of the global arrowing outcome.

The sufficient condition proved in the paper can be summarized as follows:

1. There exists an infinite family F of graphs of unbounded size.
2. For each graph G in F there are fixed-size sender S and determiner D.
3. After inserting S and D into G (producing G′), the distribution of r‑neighborhood types in G′ is identical for all G′ in the family, for any r that corresponds to the quantifier depth of a candidate FO sentence.

If these three points hold, any FO sentence attempting to capture the arrowing property will fail, because the sentence cannot see beyond the bounded radius where the structures are indistinguishable.

The authors demonstrate the feasibility of the condition with concrete constructions. For complete graphs Kₙ they use triangles (K₃) as senders and replicate them in a regular pattern to act as determiners; for cycles Cₙ and certain tree families they devise analogous gadgets. In each case they compute the necessary radius r and verify, via Hanf’s counting argument, that the r‑type distributions coincide.

A notable application is the classic arrowing statement Kₙ → (Kₘ, Kₗ). The paper shows that, although this statement is known to be FO‑non‑definable by algebraic or combinatorial arguments, the sender‑determiner framework provides a uniform proof that fits into the Hanf‑type methodology. This unifies several previously disparate non‑definability results under a single logical‑combinatorial umbrella.

Finally, the paper discusses the scope and limitations of the condition. It is not claimed to be necessary; rather, it offers a practical recipe for generating FO‑non‑definability proofs for many natural arrowing problems. The authors suggest future work on relaxing the rigidity of senders and determiners, extending the approach to richer logics such as FO+TC (first‑order with transitive closure) or monadic second‑order logic, and exploring connections with parameterized complexity. The overall contribution is a clear, technically robust bridge between finite model theory and extremal graph combinatorics, opening new avenues for analyzing the logical complexity of combinatorial decision problems.