In this expository note we present simple proofs of the lower bound of Ramsey numbers (Erdos theorem), and of the estimation of discrepancy. Neither statements nor proofs require any knowledge beyond high-school curriculum (except a minor detail). Th
us they are accessible to non-specialists, in particular, to students. Our exposition is simpler than the standard exposition because no probabilistic language is used. In order to prove the existence of a `good object we prove that the number of `bad objects is smaller than the number of all objects.
In 2019 P. Patak and M. Tancer obtained the following higher-dimensional generalization of the Heawood inequality on embeddings of graphs into surfaces. We expose this result in a short well-structured way accessible to non-specialists in the field.
Let $Delta_n^k$ be the union of $k$-dimensional faces of the $n$-dimensional simplex. Theorem. (a) If $Delta_n^k$ PL embeds into the connected sum of $g$ copies of the Cartesian product $S^ktimes S^k$ of two $k$-dimensional spheres, then $ggedfrac{n-2k}{k+2}$. (b) If $Delta_n^k$ PL embeds into a closed $(k-1)$-connected PL $2k$-manifold $M$, then $(-1)^k(chi(M)-2)gedfrac{n-2k}{k+1}$.
We present a well-structured detailed exposition of a well-known proof of the following celebrated result solving Hilberts 13th problem on superpositions. For functions of 2 variables the statement is as follows. Kolmogorov Theorem. There are conti
nuous functions $varphi_1,ldots,varphi_5 : [,0, 1,]to [,0,1,]$ such that for any continuous function $f: [,0,1,]^2tomathbb R$ there is a continuous function $h: [,0,3,]tomathbb R$ such that for any $x,yin [,0, 1,]$ we have $$f(x,y)=sumlimits_{k=1}^5 hleft(varphi_k(x)+sqrt{2},varphi_k(y)right).$$ The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.
In this survey we present applications of the ideas of complement and neighborhood in the theory embeddings of manifolds into Euclidean space (in codimension at least three). We describe how the combination of these ideas gives a reduction of embedda
bility and isotopy problems to algebraic problems. We present a more clarified exposition of the Browder-Levine theorem on realization of normal systems. Most of the survey is accessible to non-specialists in the theory of embeddings.
We present a motivated exposition of the proof of the following Tverberg Theorem: For every integers $d,r$ any $(d+1)(r-1)+1$ points in $mathbb R^d$ can be decomposed into $r$ groups such that all the $r$ convex hulls of the groups have a common poin
t. The proof is by well-known reduction to the Barany Theorem. However, our exposition is easier to grasp because additional constructions (of an embedding $mathbb R^dsubsetmathbb R^{d+1}$, of vectors $varphi_{j,i}$ and statement of the Barany Theorem) are not introduced in advance in a non-motivated way, but naturally appear in an attempt to construct the required decomposition. This attempt is based on rewriting several equalities between vectors as one equality between vectors of higher dimension.
A low-dimensional version of our main result is the following `converse of the Conway-Gordon-Sachs Theorem on intrinsic linking of the graph $K_6$ in 3-space: For any integer $z$ there are 6 points $1,2,3,4,5,6$ in 3-space, of which every two $i,j$
are joint by a polygonal line $ij$, the interior of one polygonal line is disjoint with any other polygonal line, the linking coefficient of any pair disjoint 3-cycles except for ${123,456}$ is zero, and for the exceptional pair ${123,456}$ is $2z+1$. We prove a higher-dimensional analogue, which is a `converse of a lemma by Segal-Spie.z.
This note is purely expository and is an extended version of math review to the paper [AP19]=arXiv:1901.07918v3 by S. Abramyan and T. Panov published in Proc. of Steklov Math. Inst. 305 (2019). The authors construct simplicial complexes for whose mom
ent-angle complexes certain homotopy classes are non-trivial. I present in a shorter and clearer way the main definition and the statement of Theorem 5.1 from [AP19]. The clarification reveals that the main definition used in the statements of the main results is not given [AP19].
This note is purely expository and is in Russian. We show how to prove interesting combinatorial results using the local Lovasz lemma. The note is accessible for students having basic knowledge of combinatorics; the notion of independence is defined
and the Lovasz lemma is stated and proved. Our exposition follows `Probabilistic methods of N. Alon and J. Spencer. The main difference is that we show how the proof could have been invented. The material is presented as a sequence of problems, which is peculiar not only to Zen monasteries but also to advanced mathematical education; most problems are presented with hints or solutions.
This book is expository and is in Russian (sample English translation of two pages is given). It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear different notions of curvature, which d
istinguish given geometry from the ordinary one. Direct elementary definitions of these notions are presented. The book is accessible for students familiar with analysis of several variables, and could be an interesting easy reading for professional mathematicians. The material is presented as a sequence of problems, which is peculiar not only to Zen monasteries but also to serious mathematical education.
It is presented the simplest known disproof of the Borsuk conjecture stating that if a bounded subset of n-dimensional Euclidean space contains more than n points, then the subset can be partitioned into n+1 nonempty parts of smaller diameter. The
argument is due to N. Alon and is a remarkable application of combinatorics and algebra to geometry. This note is purely expository and is accessible for students.
