# Binary matroid representation of cyclic families

Let $B=\{b_1,b_2,\ldots ,b_k\}\subset\{0,1,\ldots ,n-1\}$, and let $B_i=\{b_1+i, b_2+i,\ldots, b_k+i\}$ where addition is modulo n. That is, ${\mathcal B}=\{B_1,B_2,\ldots B_N\}$ is the collection of cyclic translates of B. Prove that there exists a binary matroid M such that every $B_i$ is a basis of M.

## Remarks

This question was asked by Attila Sali. If true, then it would prove an extremal set theoretical conjecture of Vera T. Sós, see [1].

It was proven (implicitly) for $B=\{1,2,\ldots k\}$ by Griggs and Walker [2]. In this case, there even exist graphic matroids as it was proved by Sali and Simonyi [3].

If n=7 and $B=\{1,2,4\}$, then the cyclic translates form the Fano plane, so graphic matroid cannot be expected for general B.

The statement was proven for k=3 by Füredi et al.[4].

## References

1. F.R.K. Chung, R.L. Graham, P. Frankl, J.B. Shearer, Some intersection theorems for ordered sets and graphs, J. Combin. Theory Ser. A 43 (1986), no. 1, 23-37. DOI link, Author link.
2. J.R. Griggs, J.W. Walker, Anticlusters and intersecting families of subsets, J. Combin. Theory Ser. A 51 (1989), no. 1, 99-103, DOI link.
3. A. Sali, G. Simonyi, Intersecting set systems and graphic matroids, Discrete Math. 185 (1998), 279-285, DOI link.
4. Z. Füredi, J.R. Griggs, R. Holzman, D.J. Kleitman, Representations of families of triples over GF(2), J. Combin. Theory Ser. A 53 (1990), no. 2, 306-315. DOI link, Author link.