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The aim of this short note is to present an elementary, self-contained, and direct proof for the classical Lebesgue decomposition theorem.

In this note we give a short, direct proof of the well known Combinatorial Nullstellensatz.

We give a new simpler proof of a theorem of Jayne and Rogers.

Based on various strategies, we obtain several simple proofs of the celebrated Sharkovsky cycle coexistence theorem.

In this note, we present a simple directed graph proof of Sharkovsky's theorem.

We announce here that Fermat's Last theorem was solved, but there is an easy proof of it on the basis of elemetary undergraduate mathematics. We shall disclose such an easy proof.

We give a very simple proof of a strengthened version of Chernoff's Inequality. We derive the same conclusion from much weaker assumptions.

We give a proof of the o-minimal version of the Whitney Extension Theorem simplified as compared to the original ones. A new simplifying ingredient is a definable variant of Urysohn's lemma for class $\mathcal{C}^q$ (see Section 3).

A very short proof of G\"odel's second incompleteness theorem (for set theory, second order arithmetic etc.)

We give a simple proof of the increasing strengthening of Arhangel'skii's Theorem. Our proof naturally leads to a refinement of this result of Juh\'asz.

A very short proof of the Fej\'er-Riesz lemma is presented in the matrix case

In this note we exhibit a very simple proof of McNaughton Theorem, almost right out of the definitions, and at the same time we observe that this theorem does not depend of Chang's completeness theorem.

I present a simple, elementary proof of Morley's theorem, highlighting the naturalness of this theorem.

In this note we give two proofs of Brooks' Theorem. The first is obtained by modifying an earlier proof and the second by combining two earlier proofs. We believe these proofs are easier to teach in Computer Science courses.

Simple and shorter proofs of two Dirac-type theorems involving connectivity are presented.

A short, fairly self-contained proof is given of the Poincar\'e Conjecture. In the previous version there was an error on Page 8. This gap has now been filled.

A very short proof of Kneser's theorem via transversal is given.

A simple proof of the celebrated theorem of Lee and Yang is attempted in this short note.

We give a pen and paper and (comparatively) much simpler proof to verify of the Four Colour Theorem.

We present a short and self-contained proof of the choosability version of Brooks' theorem.