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## The existence of special designs

In 2018 Mario Krenn posed this question, originated from recent advances in quantum physics. Despite very intensive Krenn’s promotion and our efforts, the question is answered only in special cases. For instance, the case of $$n=4$$ vertices was solved by kevin by means of Groebner bases. But, as far as I know, already for $$n=6$$ the answer is unknown and even a description of two-colored answers seems to be a too hard problem.

So now we are considering the case of the monochrome-edge-only graph. In this case the weight of the vertex-colored graph splits into the product of weights of its vertex-monochrome subgraphs. This reduces the problem and we were able to obtain upper bounds on a number $$C(n)$$ of colors which can have a monochrome-edge-only monochromatic graph on $$n$$ vertices. As far as I know, the present best bounds are $$C(6)=2$$, $$2\le C(8)\le 3$$and $$2\le C(n)\le n-2$$ for all even $$near 10$$.

Now I am trying to improve these bounds by means of auxiliary designs. Namely, it is easy to check that a monochrome-edge-only monochromatic graph $$G$$ with and $$K$$ colors and the vertex set $$V$$ of even order $$n\ge 6$$ induces the special design with $$n$$ vertices, that is $$K$$ families $$\mathcal V_1,\dots, \mathcal V_K$$ of nonempty even-order subsets of $$V$$ satisfying the following conditions:

1. $$V\in\mathcal V_i$$ for each $$i\in\{1,\dots,K\}$$.

2. For each partition of $$V$$ into $$m>1$$ even-order non-empty subsets $$V_1,\dots, V_m$$ and each distinct indices $$i_1,\dots, i_m\in\{1,\dots,K\}$$ there exists an index $$i_j$$ look for that $$V_i\not\in \mathcal V_{i_j}$$.

3. let $$V’$$ be any even order $$>2$$ subset of the set $$V$$, $$v$$ be any vertex of $$V’$$and $$i\in\{1,\dots,K\}$$ be any index. If $$V’\in\mathcal V_i$$ then there exists a vertex $$u\in V’\setminus\{v\}$$ such that both sets $$\{v,u\}$$ other $$V’\setminus\{v,u\}$$ belong to $$\mathcal V_i$$.

For instance, the known examples of monochrome-edge-only monochromatic graphs with $$n$$ vertices and $$2$$ colors provide the special design on the set $$V=\{1,\dots,n\}$$ with $$\mathcal V_1$$ consisting of all nonempty unions of the doubletons $$\{1,2\},\{3,4\},\dots, \{n-1,n\}$$ other $$\mathcal V_2$$ consisting of all nonempty unions of the doubletons $$\{2,3\},\{4,5\},\dots, \{n-2,n-1\}, \{n,1\}$$.

Thus $$C(n) ≤ K(n)$$where $$K(n)$$ is the maximally positive integer $$K$$ such that there exists a special design with $$n$$ vertices and $$K$$ families, and so I want to estimate $$K(n)$$. Unfortunately, the known asymptotic bounds on $$K(n)$$ are so loose that I do not even know whether $$K(n)$$ or $$nK(n)$$ is $$o(n)$$. Finally I note that it is easy to show that for each even number $$n\ge 6$$ holds $$K(n)where k(n) is a rather natural graph-theoretical characteristic, introduced in this question. Unfortunately, the known asymptotic bounds for k(n) are loose too, so they only provide bounds K(6)\le 3 other K(n)\le n-2 for each n\ge 8.$$

thanks

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