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For any finite-dimensional vector space ${\mathcal F}$ of continuousfunctions $f:{\mathbb R}^1 \to {\mathbb R}^1$ consider subspaces in ${\mathcalF}$ defined by systems of {\em equality conditions} $f(a_i) = f(b_i)$, where$(a_i, b_i)$, $i=1, \dots, n$, are some pairs of points in ${\mathbb R}^1$. Itis proved that if $\dim {\mathcal F} < 2n-I(n)$, where $I(n)$ is the number ofones in the binary notation of $n$, then there necessarily are independentsystems of $n$ equality conditions defining the subspaces of codimensiongreater than $n$ in ${\mathcal F}$. We also prove lower estimates of the sizesof the inevitable drops of the codimensions of these subspaces.  These estimates are then applied to knot theory (in which systems of equalityconditions are known as {\em chord diagrams}). The inevitable presence ofcomplicated non-stable terms in sequences of spectral sequences calculating thecohomology groups of spaces of knots is proved.  Keywords: Chord diagram, configuration space, characteristic class

Varieties of chord diagrams, braid group cohomology and degeneration of equality conditions