Related papers: On the infinitary van der Waerden Theorem

We give a purely combinatorial proof for a two-fold generalization of van der Waerden-Brauer's theorem and Hindman's theorem. We also give tower bounds for a finite version of it.

We present a short new proof of the canonical polynomial van der Waerden theorem, recently established by Girao [arXiv:2004.07766].

We prove an infinitary version of the Brauer-Schur theorem.

The famous van der Waerden theorem states that if partition N into finitely many cells then one of them will contain arbitrary length arithmetic progressions. It has a polynomial version also. In this article we will prove the near 0…

In this article, we investigate polynomial generalizations of the van der Waerden theorem with a focus on largeness properties of recurrence patterns. We prove an $IP_r^\star$-strengthened version of the polynomial van der Waerden theorem,…

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 prove a canonical polynomial Van der Waerden's Theorem. More precisely, we show the following. Let $\{p_1(x),\ldots,p_k(x)\}$ be a set of polynomials such that $p_i(x)\in \mathbb{Z}[x]$ and $p_i(0)=0$, for every $i\in \{1,\ldots,k\}$.…

We present a new short proof of Van der Waerden's Theorem about the existence of arbitrarily long monochromatic arithmetic progressions. The proof uses algebra in the compact space of ultrafilters $\beta\N$, but contrarily to the other…

We present a self-contained proof of a strong version of van der Waerden's Theorem. By using translation invariant filters that are maximal with respect to inclusion, a simple inductive argument shows the existence of "piecewise…

We present an elementary combinatorial proof of the celebrated Friendship theorem. The proof involves looking at independent sets and constructing a bound on their size which forces a contradiction.

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

We present a proof of the sufficiency of Rado's condition for the partition regularity of linear Diophantine equations that avoids any use of van der Waerden's theorem. The proof is based on fundamental properties that are common knowledge…

We give a proof of an infinitary version of the well known Hales-Jewett theorem on finite words avoiding the use of ultrafilters.

We give a one-sentence elementary proof of the combinatorial Fa\`a di Bruno's formula.

The Dense Hindman's Theorem states that, in any finite coloring of the integers, one may find a single color and a "dense" set $B_1$, for each $b_1\in B_1$ a "dense" set $B_2^{b_1}$ (depending on $b_1$), for each $b_2\in B_2^{b_1}$ a…

We provide a combinatorial proof of an infinite extension of the Hales--Jewett theorem due to T. Carlson and independently due to H. Furstenberg and Y. Katznelson

We prove a combination theorem for PD(n)-pairs.

We provide a simple proof of Kamp's theorem.

The paper is devoted to a somewhat idiosyncratic account of the theorem of de Bruijn-Erd\"{o}s and Hanani from the combinatorics of finite geometries and its various proofs. Among the proofs discussed are the original proofs by de…

We give an elementary, self-contained, and purely combinatorial proof of the Rayleigh monotonicity property of graphs.