Fun Problems

I have listed a small subset of my favourite (🤔) mathematical statements (can be true/false/open/famous)! Maybe give some thought if you like any 🙂. Also, here is a small collection of some nice puzzles. Have fun!

Statement 1: There is a set of points in the plane such that every line intersects the set in exactly two points.
Statement 2: There is a dense subset D of the plane (R^2) such that the Euclidean distance between any two points of D is rational. (A subset D is dense in R^2 if and only if every disk in R^2 contains a point of D.)
Statement 3: Let S be an infinite set of natural numbers and let P(S) denote the set of all primes p which divides some number in S. Also let S+1 denote the set containing all natural numbers of the form s+1 where s belongs to S. Now if P(S) has finite cardinality then P(S+1) has infinite cardinality.
Statement 4: There exists a subset S of real numbers such that both the difference sets S-S and S'-S' do not contain any open interval. (Here S' denotes the complement of the set S and the difference set of a subset A of real numbers, denoted by A-A, consists of all elements of the form (a_1-a_2) where a_1, a_2 belongs to A.)
Statement 5: Any two graphs on at least four vertices with the same sets of vertex-deleted subgraphs are isomorphic.
Statement 6: For every finite union-closed family of sets, other than the family containing only the empty set, there exists an element that belongs to at least half of the sets in the family. ( A family of sets is said to be union-closed if the union of any two sets from the family belongs to the family. )
Statement 7: Let w_1, w_2, ...., w_n be points on the unit circle (boundary of the unit disk) in the plane. Then there exists a point z on the unit circle such that the product of the Euclidean distances from z to the points w_j, 1≤ j ≤ n, is at least 1.
Statement 8: Every graph with its chromatic number equal to k has a k-vertex complete graph as a minor.
Statement 9: Let S={G_1, G_2, G_3, ......} be any infinite set of finite simple graphs. Then there exists G_i, G_j in S such that G_i is a minor of G_j.
Statement 10: There exists an acyclic orientation of any triangle-free planar graph such that, there exists at most one directed path between any two of its vertices.
Statement 11: Given any point set P containing n points inside the unit disk, let G(P) denote the Euclidean minimum spanning tree of P. Let e_1, e_2, e_3, ....e_{n-1} denote the lengths of the n-1 edges of G(P). Then the sum of the squares of the edge lengths of G(P) i.e (e_1)^2+ (e_2)^2+ (e_3)^2+.....+ (e_{n-1})^2 is bounded by a constant independent of n. (What if instead of squares we take the 3/2th powers?)
Statement 12: Suppose one colors the vertices of any planar graph P with two colors red and blue in any way such that the number of red vertices is twice as large as the number of blue vertices. Then there is a subset S of the red vertices of P such that i) the cardinality of S is bounded by a constant (independent of the graph P) and ii) the number of blue vertices adjacent to some vertex of S is strictly less than the cardinality of S.
Statement 13: In any simple directed graph, there exists a vertex v whose second neighborhood (all vertices u which are out-neighbor of some out-neighbor of v) is at least as large as its first neighborhood (out-neighbors of v).
Statement 14: Let A be an infinite subset of the natural numbers such that the sum of the reciprocals of the numbers in A is infinity. Then A contains arithmetic progressions of any arbitrary length.

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