LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

If $m \to (s - 1, t)^{2}$ and $n \to (s, t - 1)^{2}$ , then $m + n \to (s, t)^{2}$ for $s, t \geq 2$

Statement

Let $s, t \geq 2$ . If $m \to (s - 1, t)^{2}$ and $n \to (s, t - 1)^{2}$ , then $m + n \to (s, t)^{2}$ in the notation of Finite colourings of $k$ -element subsets, monochromatic sets, and the arrow notations $N \to (s, t)^{2}$ and $N \to (r)_{c}^{k}$ . Finite sums and cardinalities use The sum rule: a finite disjoint union is finite with $∣ A \cup B ∣ = ∣ A ∣ + ∣ B ∣$ and $∣ ⋃_{i \in I} A_{i} ∣ = \sum_{i \in I} ∣ A_{i} ∣$ , and a sum over a finite index set splits along a partition and The cardinality $∣ A ∣$ of a finite set, and complete graphs use Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices.

Facts & Assumptions

Given: Naturals $m, n$ and $s, t \geq 2$ satisfying the two displayed arrow hypotheses, and an arbitrary red-blue colouring of the pairs of an $(m + n)$ -element vertex set.

[F1]

A red-blue colouring witnesses $N \to (s, t)^{2}$ when it contains a red $s$ -set or a blue $t$ -set (Finite colourings of $k$ -element subsets, monochromatic sets, and the arrow notations $N \to (s, t)^{2}$ and $N \to (r)_{c}^{k}$ ).

Proof

technique · direct

1.1

Fix a vertex $v$ . Partition the other $m + n - 1$ vertices into the red neighbours $A$ of $v$ and the blue neighbours $B$ of $v$ . If $∣ A ∣ \geq m$ , restrict to an $m$ -element subset of $A$ and apply $m \to (s - 1, t)^{2}$ ; if $∣ A ∣ < m$ , then $∣ B ∣ \geq n$ by the finite sum rule, so restrict to an $n$ -element subset of $B$ and apply $n \to (s, t - 1)^{2}$ .

givenF1

2.1

In the first case, a red $(s - 1)$ -set in $A$ becomes a red $s$ -set after adjoining $v$ , while a blue $t$ -set already works. In the second case, a blue $(t - 1)$ -set in $B$ becomes a blue $t$ -set after adjoining $v$ , while a red $s$ -set already works. Hence every colouring has one of the alternatives in [F1], so $m + n \to (s, t)^{2}$ .

step 1.1F1∎

Depends on

Used by

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Fox et al., Graph Ramsey Theory, Section 2.1 (standard reference, not scraped)
R. Diestel, Graph Theory, 6th ed., Chapter 9, Section 9.1 (standard reference, not scraped)

If m→(s−1,t)2m→(s−1,t)2 and n→(s,t−1)2n→(s,t−1)2, then m+n→(s,t)2m+n→(s,t)2 for s,t≥2s,t≥2