Recently I attacked [this](https://mathoverflow.net/questions/363026/what-is-the-best-way-to-partition-the-4-subsets-of-1-2-3-dots-n/363104#363104) combinatorial question. The value of $m(n)$ introduced in it equals to a [clique cover number](https://en.wikipedia.org/wiki/Clique_cover)
of a [generalized Kneser graph](https://en.wikipedia.org/wiki/Kneser_graph#Related_graphs) $KG_{n,4,1}=K(n,4,2)$ (or the chromatic number of its complement). I tried to find references with bounds for this number, but failed.  On the other hand, the chromatic number of generalized Kneser graphs was investigated, see the references. For instance, if $n = (k-1)s+r$, $0\le r<k–1$ then Proposition 2.6 from [AJ] implies that $\chi(K(n,k,2))\le (k-1){s\choose 2}+rs$ and Frankl [F] showed that if $n >10k^3e^k$ then this bound is exact. 

RobPratt [calculated](https://math.stackexchange.com/questions/3717109/what-is-the-best-way-to-partition-the-4-subsets-of-1-2-3-dots-n/3717979#3717979) the values of $m(n)$ for $m\le 9$. Max Alekseyev [showed](https://mathoverflow.net/questions/363026/what-is-the-best-way-to-partition-the-4-subsets-of-1-2-3-dots-n/363104#comment915739_363026) that $m(n)\ge \frac{(n-2)(n-3)}2$. If $n$ is a power of an odd prime using a finite field of order $n$ we can [show](https://math.stackexchange.com/questions/3717109/what-is-the-best-way-to-partition-the-4-subsets-of-1-2-3-dots-n/3721079#3721079) that $m(n)\le n^2$.  This observation implies an upper bound $m(n)\le (n+n^{0.525})^2$ for sufficiently large $n$, becase for sufficiently large $x$ there is a prime belonging to $[x-x^{0.525}, x]$, see [BHP]. 

Thanks. 

*References*

[AJ] Sharareh Alipour, Amir Jafari, *[On the chromatic number of generalized Kneser graphs](https://cdm.ucalgary.ca/article/download/62380/46789/)*, Contributions to discrete mathematics **12**:2 69–76.

[BCK] József Balogh, Danila Cherkashin, Sergei Kiselev, *[Coloring general Kneser graphs and hypergraphs via high-discrepancy hypergraphs](http://arxiv.org/abs/1805.09394)*.

[BHP] R. Baker, G. Harman, J. Pintz, *[The difference between consecutive primes. II](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.360.3671&rep=rep1&type=pdf)*.
Proc. Lond. Math. Soc., (3) Ser. 83 (2001) 532–562.

[F] Peter Frankl, *On the chromatic number of the general Kneser-graph*, Journal of Graph Theory, **9**:2 (1985) 217–220.

[FF] Peter Frankl, Zoltán Füredi, *Extremal problems concerning Kneser graphs*, Journal of Combinatorial Theory, Series B, **40**:3 (1986) 270–284.