Corollary. A refinement by projecting the ambient space [cn-0074]

    Status:

    Statement. Let \(m\geq 1\), let \(W\) be a finite-dimensional real vector space, and let \(U_i,V_i\subseteq W\), \(1\leq i\leq m\), satisfy \(U_i\cap V_i=0\). Put

    \[r=\max _{1\leq i\leq m}\bigl (\dim U_i+\dim V_i\bigr ).\]

    Then the subspace-intersection bound [cn-0024] has the refinement

    \[\sum _{i,j=1}^{m}\dim (U_i\cap V_j)\leq \left \lfloor \frac {m^2}{4}\right \rfloor r.\]

    Proof. The case \(r=0\) is immediate. Write \(C_i=U_i+V_i\); diagonal disjointness gives \(\dim C_i\leq r\). There is a linear map \(\pi :W\to \mathbb R^r\) injective on every \(C_i\), by Scott and Wilmer, Lemma 3.1 [cn-0071]. If \(r=\dim W\), take an isomorphism; otherwise all the \(C_i\) are proper subspaces, as required there.

    Here is the needed existence argument directly. Regard the entries of an \(r\times \dim W\) matrix as variables. For each nonzero \(C_i\), choose a basis and a maximal minor of its image matrix that is a nonzero polynomial: such a minor exists because some linear map embeds \(C_i\) into \(\mathbb R^r\). The product of these finitely many nonzero polynomials is nonzero, so it is nonzero at some real matrix. That matrix defines the required \(\pi \). Zero spaces impose no condition.

    Injectivity on \(C_i\) gives \(\pi (U_i)\cap \pi (V_i)=0\). For every \(i,j\), the map is injective on \(U_i\cap V_j\subseteq C_i\), and

    \[\pi (U_i\cap V_j)\subseteq \pi (U_i)\cap \pi (V_j).\]

    Apply the subspace inequality [cn-0024] to the image pairs in \(\mathbb R^r\):

    \[\sum _{i,j}\dim (U_i\cap V_j)\leq \sum _{i,j}\dim \bigl (\pi (U_i)\cap \pi (V_j)\bigr )\leq \left \lfloor \frac {m^2}{4}\right \rfloor r.\]

    The bound is sharp: place the balanced equality construction [cn-0025] inside a fixed \(r\)-dimensional subspace of \(W\). The argument improves the parameter in the bound when \(r<\dim W\); it still relies on the general inequality in dimension \(r\). For the graph subspaces in the crossing proof [cn-0072], one has \(r=\dim W=2d\).

    Proof dependencies
    Subspace inequality [cn-0024]; Sharpness construction [cn-0025]; Scott–Wilmer projection lemma [cn-0071]
    Provenance
    Editorial consequence of OpenAI [cn-0042], Lemma 2.1, and the generic-projection argument of Scott and Wilmer [cn-0071], arXiv:1907.06019v2, Lemma 3.1. Added on 12 October 2026, with the needed projection argument written out below.