Content disclaimer: It's really hard for me to judge the difficulty of these problems. They all have an easy-to-understand appearance but may be very difficult indeed, and it's very easy to become addicted to them. So try them at your own risk!

**1.
Erdős–Faber–Lovász
conjecture**

The edge disjoint union of k cliques of size k has chromatic number k.

**2. Hedetniemi
conjecture **

Given two graphs G and H, define the graph tensor product G*H so that the adjacency matrix of the product graph is equal to the tensor product of two adjacencies matrices. The conjecture is: the chromatic number of G*H is equal to the minimum of the two chromatic numbers.

**3. Kusner
conjecture **

**There
are at most 2n points
in ℝ**^{n},
such that the pairwise L_{1} distances between any two points
are equal to 1.

_{4.
Revolving door conjecture (Havel)}

All the k-subsets and (k+1)-subsets of [2k+1] can be listed on a cycle by alternately adding or deleting one element.

_{5.
Bunkbed
conjecture (Kasteleyn)}

Given a graph G, make another copy of it and connect all the corresponding vertices (say u and u', v and v' and etc.) by "vertical edges" (think about setting up a bunkbed). Now take every edge independently in some probability such that the probability of taking uv is same as that of u'v'. Then the probability that there exists a path from u to v is no less than the probability that there exists a path from u to v'.

_{6. Unit
distance problem (Erd}_{ős}_{)}

Among n points on the plane, how many pairs of points can be at unit distance from each other?

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