= Mod-two diagonal class of the real projective plane
{title2=$\operatorname{PD}_{\mathbb F_2}(\Delta)=x^2+xy+y^2$}
The <cohomology ring> of the product of two <real projective planes> is $\mathbb F_2[x,y]/(x^3,y^3)$, with $|x|=|y|=1$. Pairing against $x^2$, $xy$ and $y^2$ identifies the <Poincare dual> of the diagonal as $x^2+xy+y^2$. For a self-map $f$ with $f^*x=\varepsilon x$, the graph pullback evaluates on the mod-two <fundamental class> as $1+\varepsilon+\varepsilon^2=1$. Thus no graph of a map on the whole projective plane can avoid the diagonal. The argument uses a relative <Thom class> and does not assume transverse intersections.
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