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These two four below problems are interesting and still open conjecture in design theory and its related topics:

$1.$ There exist Ucycles for $k$-subsets of $[n]$, provided $k$ divides $Cr(n-1,k-1)$ and $n$ is sufficiently large.

$2.$ For $n≥6$ even, it is not possible to have a length $\frac{n^2}{2}$ cyclic covering word for the $(n−2)$-subsets of an $n$-set.

$3.$ For each $k≥3$, there exists a constant $c_k$ such that for all $v≥c_k$ and $λ≥1$, the $1$ block-intersection graph of any $BIBD(v,k,λ)$ is Hamiltonian.

$4.$ For each $k≥3$, there exists a constant $c_k$ such that for all $v≥c_k$ and $λ≥1$, the ${1,2}$-block-intersection graph of any $BIBD(v,k,λ)$ is Hamiltonian.

These two four conjectures are respectively from:

$1.$ Chung, F., Diaconis, P., Graham, R.: Universal cycles for combinatorial structures. Discrete Math.110, 43–59 (1992)

$2.$ Stevens, B., Buskell, P., Ecimovic, P., Ivanescu, C., Malik, A., Savu, A., Vassilev, T., Verrall, H., Yang, B., Zhao, Z.: Solution of an outstanding conjecture: the non-existence of universal cycles withk=n−2. Discrete Math.258, 193–204 (2002)

Also

$3.$ Jesso, Andrew T.(3-NF); Pike, David A.(3-NF); Shalaby, Nabil(3-NF) Hamilton cycles in restricted block-intersection graphs. (English summary) Des. Codes Cryptogr.61(3), 345–353

$Also,$ I belive believe that ; Hadamard conjecture is a nice diamond in the conjectures land.

1

These two below problems are interesting and still open conjecture in design theory and its related topics:

$1.$ There exist Ucycles for $k$-subsets of $[n]$, provided $k$ divides $Cr(n-1,k-1)$ and $n$ is sufficiently large.

$2.$ For $n≥6$ even, it is not possible to have a length $\frac{n^2}{2}$ cyclic covering word for the $(n−2)$-subsets of an $n$-set.

These two conjectures are respectively from:

$1.$ Chung, F., Diaconis, P., Graham, R.: Universal cycles for combinatorial structures. Discrete Math.110, 43–59 (1992)

$2.$ Stevens, B., Buskell, P., Ecimovic, P., Ivanescu, C., Malik, A., Savu, A., Vassilev, T., Verrall, H., Yang, B., Zhao, Z.: Solution of an outstanding conjecture: the non-existence of universal cycles withk=n−2. Discrete Math.258, 193–204 (2002)

Also I belive that; Hadamard conjecture is a nice diamond in the conjectures land.