An inequality in binary vector spaces

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A. E. Brouwer, An inequality in binary vector spaces, Discrete Mathematics 59 (1986), no. 3, 315–317. The edition read is the three-page journal facsimile supplied by the CWI repository, at its PDF address, accessed 2026-10-16.

Theorem (printed page 315)

In the finite-dimensional setting, let $X$ be a vector space over $\mathbb F_2$, let $V,U_1,\ldots,U_k$ be linear subspaces of $X$, and let $a_1,\ldots,a_k\in X$. Suppose the cosets $a_i+U_i$ cover $V$, and deleting any one coset leaves some point of $V$ uncovered. Set

$$W=V\cap\bigcap_{i=1}^{k}U_i.$$

Then $\dim V-\dim W\leq k-1$. In particular, an irredundant cover of a binary vector space by $k$ affine subspaces has common direction-space intersection of codimension at most $k-1$. Irredundancy is relative to the covered space; it cannot be dropped while retaining the same conclusion.

Read status: claims checked. The statement's field, cover condition, irredundancy and common-intersection definition were checked against the printed page. The paper's proof occupies pages 315–316; it has not been independently verified here. The remarks on other finite fields are outside this card's consumed scope.

Bears on. The theorem supplies the affine-cover premise for Proposition 8's bound on binary coset-fibre constructions toward Problem 483. The application restricts to an affine half, takes an irredundant subcover, and intersects its directions to remove colors. That native deduction is author-recorded and unreviewed; the source theorem itself gives no bound for general Schur numbers.

No file of this source is held: no license on record permits its redistribution. The journal facsimile's printed page 315 carries Elsevier Science Publishers' 1986 copyright notice. The CWI record read on 2026-10-16 names no open license for this edition, so the recorded term is reserved.