\input zb-basic \input zb-ioport \iteman{io-port 04156465} \itemau{Heinrich, Katherine} \itemti{Coloring the edges of $K\sb m\times K\sb m$.} \itemso{J. Graph Theory 14, No.5, 575-583 (1990).} \itemab For any two positive integers $\ell$ and c, f($\ell,c)$ is the least integer k such that if $\{d\sb j:$ $1\le j\le \ell \}$ are c-colourings of the set $$V=\{(c\sb 1,c\sb 2,...,c\sb{2\ell -1}):\ c\sb i\in \{1,2,...,k\}\},$$ then, for each $j\in \{1,2,...,\ell \}$, there exist integers $a\sb j$ and $b\sb j$, $1\le a\sb j\le b\sb j\le k$, so that $$d\sb j((a\sb 1,b\sb 1,a\sb 2,b\sb 2,...,a\sb{j-1},b\sb{j-1},a\sb j,a\sb{j+1},b\sb{j+1},...,a\sb{\ell},b\sb{\ell}))=$$ $$d\sb j((a\sb 1,b\sb 1,a\sb 2,b\sb 2,...,a\sb{j-1},b\sb{j-1},b\sb j,a\sb{j+1},b\sb{j+1},...,a\sb{\ell},b\sb{\ell})).$$ Any improvement in the upper bounds on f($\ell,c)$ yields immediately an improvement in the upper bounds on the van der Waerden numbers. The main result is the following: There exists a constant k such that $f(2,c)>kc\sp 3$. To prove this c-colourings of the edges of $K\sb m\times K\sb m$ are considered. \itemrv{U.Baumann} \itemcc{} \itemut{van der Waerden numbers; c-colourings} \itemli{doi:10.1002/jgt.3190140509} \end