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 for the general audience who loves to think. Have fun!
Thinking in the Age of AI (28th July, 2026):
In these uncertain times for the whole mathematical community where AI is getting better everyday and may surpass human intelligence and go far beyond, it is more important than ever that we do not forget the beauty of thinking and mathematical discovery. For this reason, humans—especially the younger generation—should engage even more deeply in brainstorming, exploration, and the joy of independent thought. In an age when machines may be able to produce thoughts faster than us, the human act of appreciating a thought becomes even more precious. For understanding is not merely receiving an answer, but recognizing its beauty, meaning, and place in the larger landscape of ideas.
Mathematics is the language of the universe! But if we need a translator at every step, do we ever truly hear or understand what it says?Â
Statement 1:Â 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: Â 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: Â 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 Â 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.Â
Statement 13: Â 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: Â 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.Â
     Â