Related papers: The Tur\'{a}n number and probabilistic combinatori…
In this note, we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic…
We show the existence of regular combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, t-designs, and t-wise…
We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, $t$-designs, and $t$-wise…
The method of Tur\'an in establishing the normal order for the number of prime divisors of a number is used to show that a certain class of arithmetic functions do not have a normal order.
Let $LA_{n}(\tau)$ be the length of the longest alternating subsequence of a uniform random permutation $\tau\in[n]$. Classical probabilistic arguments are used to rederive the asymptotic mean, variance and limiting law of $LA_{n}(\tau)$.…
The aim of this article is to promote the use of probabilistic methods in the study of problems in mathematical general relativity. Two new and simple singularity theorems, whose features are different from the classical singularity…
The article gives a survey of mathematical proofs that rely on computer calculations and formal proofs.
Motivated by a Gan-Loh-Sudakov-type problem, we introduce the regular Tur\'an numbers, a natural variation on the classical Tur\'an numbers for which the host graph is required to be regular. Among other results, we prove a striking…
In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or…
In this article we demonstrate how algorithmic probability theory is applied to situations that involve uncertainty. When people are unsure of their model of reality, then the outcome they observe will cause them to update their beliefs. We…
We introduce in this section an Algebraic and Combinatorial approach to the theory of Numbers. The approach rests on the observation that numbers can be identified with familiar combinatorial objects namely rooted trees, which we shall here…
We describe a technique for mechanically proving certain kinds of theorems in combinatorics on words, using automata and a package for manipulating them. We illustrate our technique by solving, purely mechanically, an open problem of Currie…
Recently, Domagoj Brada\v{c}, Oliver Janzer, Benny Sudakov and Istv\'an Tomon have proved that the Tur\'an number of $2$-dimensional grids is $\Theta(n^{3/2})$, or more general, $\mathrm{ex}\left(n,T\square{P}\right)=\Theta(n^{3/2})$, where…
We question the commonly accepted statement that random numbers certified by Bell's theorem carry some special sort of randomness, so to say, quantum randomness or intrinsic randomness. We show that such numbers can be easily generated by…
Alongside the development of quantum algorithms and quantum complexity theory in recent years, quantum techniques have also proved instrumental in obtaining results in classical (non-quantum) areas. In this paper we survey these results and…
Turan's method, as expressed in his books, is a careful study of trigonometric polynomials from different points of view. The present article starts from a problem asked by Turan: how to construct a sequence of real numbers x(j) (j=…
We give a short proof of Stein's universal multiplier theorem, purely by probabilistic methods, thus avoiding any use of harmonic analysis techniques (complex interpolation or transference methods).
Trough the classical umbral calculus, we provide new, compact and easy to handle expressions of k-statistics, and more in general of U-statistics. In addition such a symbolic method can be naturally extended to multivariate case and to…
A combinatorial methods are used to investigate some properties of certain generalized Stirling numbers, including explicit formula and recurrence relations. Furthermore, an expression of these numbers with symmetric function is deduced.
By presenting the proofs of a few sample results, we introduce the reader to the use of nonstandard analysis in aspects of combinatorics of numbers.