Schur–Brauer version of van der Waerden's theorem for semimodules

MathDB #332161: a two-element counterexample

Result

The stated Schur--Brauer conjecture for semimodules is false. A counterexample is the regular one-dimensional module over the field $\mathbb F_2$, split into its two singleton color classes.

The source statement

Conjecture 3.26 of Xiongping Dai, “Grünwald version of van der Waerden's theorem for semi-modules,” arXiv:1512.08695v4, says the following. If

$$M=B_1\cup\cdots\cup B_q$$

is any finite partition of a semimodule $M$ over a semiring $R$, then some fixed cell $B_j$ has the property that, for every finite $F\subseteq R$, there are $a\in M$ and $b\in B_j\setminus\{\boldsymbol o\}$ such that

$$a+Fb\subseteq B_j. \tag{1}$$

Here $Fb=\{fb:f\in F\}$. The source does not require $R$ or $M$ to be infinite, cancellative, torsion-free, or a $*$-semiring, and it places no restriction on the finite subset $F$ beyond $F\subseteq R$.

Counterexample

Take

$$R=M=\mathbb F_2=\{0,1\}$$

with the usual operations, and let $M$ be the regular left $R$-module. Partition it as

$$B_1=\{0\},\qquad B_2=\{1\}.$$

Choose the finite set

$$F=R=\{0,1\}.$$

The cell $B_1$ cannot satisfy (1), because it contains no permitted element $b\ne\boldsymbol o=0$.

For $B_2$, the only possible $b$ is $b=1$. For either $a\in M$,

$$a+Fb=\{a+0\cdot1,a+1\cdot1\}=\{a,a+1\}=\mathbb F_2,$$

which is not contained in the singleton $B_2$. Thus the same finite set $F$ defeats both color classes. No cell has the asserted property, so the conjecture is false.

Hypothesis check

The source defines a semiring by requiring an abelian additive semigroup with zero, an associative multiplicative semigroup with unit and absorbing zero, and both distributive laws. It defines a left semimodule as an abelian additive semigroup with zero and an action satisfying

$$(r+t)g=rg+tg,\qquad r(g+h)=rg+rh,\qquad 1g=g,\qquad 0g=\boldsymbol o.$$

The field $\mathbb F_2$ and its regular module satisfy every one of these axioms (as well as the usual scalar-associativity axiom). A finite algebra also causes no issue with the paper's use of the discrete topology.

There is likewise no quantifier ambiguity. The conjecture asserts

$$\exists j\ \forall F\subseteq_{\rm fin}R\ \exists a\in M\ \exists b\in B_j\setminus\{\boldsymbol o\}:a+Fb\subseteq B_j.$$

The construction proves its negation using one common choice $F=\{0,1\}$ for both cells. Although the source only requires $a\in M$, the presence of $0\in F$ would force $a=a+0b\in B_j$ whenever (1) held, so allowing $a$ outside the cell cannot rescue the claim.

Exact verification

The accompanying certificate.json records the two operation tables, the regular scalar action, the partition, and $F$. The standard-library script verify_counterexample.py independently checks all finite semiring and semimodule identities, the partition axioms, and every possible pair $(a,b)$ in the conjecture's conclusion. Run

python verify_counterexample.py

from this directory. All checks use explicit exceptions and therefore remain active under python -O.

Scope

This refutes the conjecture exactly as stated in all arXiv versions and in MathDB #332161. It does not decide a repaired version restricted to an infinite or otherwise nonperiodic class of semirings/semimodules. Any such repair needs new hypotheses that explicitly exclude this finite-field obstruction.