# Algebraic Kneser conjecture?

Recall that Kneser conjecture (now Lovasz theorem) claims that if the family of $k$-subsets (subsets of cardinality $k$) of given $(2k+d)$-set $M$, $d\geq 1$ are colored into $d+1$ colors, then there are two disjoint $k$-subsets of the same color (easy to see that for $d+2$ colors it does not hold, say, if $M=\{1,2,\dots,2k+d\}$, then color $K\subset M$ to color $i\in \{1,2,\dots,d+1\}$ if $\min K=i$ and color $K$ to color $d+2$ if $\min K>d+1$).

This may be reformulated as follows (mad on first glance) way. Consider the (say, complex) non-commutative algebra $A$ with generators $g_1$, $g_2$, $\dots$, $g_{2k+d}$ and relations $g_ixg_i=0$ for each $i=1,2,\dots,2k+d$ and each $x\in A$. This algebra is naturally graded by degree of monomials in $g_i$'s. Denote by $A_k$ homogeneous component of degree $k$. Note that if $A_k\ni x=\sum_{K} c(K)g(K)$ (here $K$ runs over $k$-subsets of index set $\{1,2,\dots,2k+d\}$, $g(K):=g_{a}g_{b}g_{c}\dots$ for $K=\{ a < b < c\dots \}$, $C(K)$ are some complex coefficients), then $x^2=0$ iff no two $K$'s with $c(K)\ne 0$ are disjoint. So, the Kneser conjecture is equivalent to the following statement:

not any element $A_k$ is a some of $d+1$ square roots of 0.

This is nothing but tautology, but the question is whether this statement holds aswell for some factors of $G$, which have nicer algebraic structure? For example, for commutative algebra with relations $g_i^2=0$, or for exterior algebra? (This last may happen only for even $k$, of course, else each element is a square root of 0).

-