New upper bound for multicolor Ramsey number of odd cycles

Let $r_k\left(C_{2m+1}\right)$ be the $k$-color Ramsey number of an odd cycle $C_{2m+1}$ of length $2m+1$. It is shown that for each fixed $m\ge 2$, $r_k\left(C_{2m+1}\right)<c^k\sqrt{k!}$for all sufficiently large $k$, where $c=c\left(m\right)>0$ is a constant. This improves an old result by Bondy and Erd?s (1973).     

Monochromatic subgraphs in randomly colored graphons

Let $T(H,G_n)$ be the number of monochromatic copies of a fixed connected graph $H$ in a uniformly random coloring of the vertices of the graph $G_n$. In this paper we give a complete characterization of the limiting distribution of $T(H,G_n)$, when ${\left\{G_n\right\}}_{n\ge 1}$ is a converging sequence of dense graphs. When the number of colors grows to infinity, depending on whether the expected value remains bounded, $T(H,G_n)$ either converges to a finite linear combination of independent Poisson variables or a normal distribution. On the other hand, when the number of colors is fixed, $T(H,G_n)$ converges to a (possibly infinite) linear combination of independent centered chi-squared random variables. This generalizes the classical birthday problem, which involves understanding the asymptotics of $T(K_s,K_n)$, the number of monochromatic $s$-cliques in a complete graph $K_n$ ($s$-matching birthdays among a group of $n$ friends), to general monochromatic subgraphs in a network.     

Anzahl theorems of matrices over the residue class ring modulo psqt and their applications

Let $h=p^s{q}^t$ be its decomposition into a product of powers of two distinct primes, and ${\mathbb{Z}}_h$ be the residue class ring modulo h. In this paper, we determine the Smith normal forms of matrices, compute the number of the orbits of $m\times n$ matrices under the action of the group $G{L}_m({\mathbb{Z}}_h)\times G{L}_n({\mathbb{Z}}_h)$ and the length of each orbit over ${\mathbb{Z}}_h$. As applications, we determine the clique number, the independence number and the chromatic number of the bilinear forms graph, construct a family of association schemes by using $1\times n$ matrices and compute the intersection numbers and the character table over ${\mathbb{Z}}_h$ for $s=t=1$.     

On invariant fields of vectors and covectors

Let ${\mathbb{F}}_q$ be the finite field of order q. Let G be one of the three groups $\mathrm{GL}(n,{\mathbb{F}}_q)$, $\mathrm{SL}(n,{\mathbb{F}}_q)$ or $\mathrm{U}(n,{\mathbb{F}}_q)$ and let W be the standard n-dimensional representation of G. For non-negative integers m and d we let $mW\oplus d{W}^{?}$ denote the representation of G given by the direct sum of m vectors and d covectors. We exhibit a minimal set of homogeneous invariant polynomials $\{{?}_1,{?}_2,\dots ,{?}_{(m+d)n}\}?{\mathbb{F}}_q{[mW\oplus d{W}^{?}]}^G$ such that ${\mathbb{F}}_q{(mW\oplus d{W}^{?})}^G={\mathbb{F}}_q({?}_1,{?}_2,\dots ,{?}_{(m+d)n})$ for all cases except when $md=0$ and $G=\mathrm{GL}(n,{\mathbb{F}}_q)$ or $\mathrm{SL}(n,{\mathbb{F}}_q)$.     

On the local geometry of graphs in terms of their spectra

In this paper we consider the relation between the spectrum and the number of short cycles in large graphs. Suppose $G_1,G_2,G_3,\dots$ is a sequence of finite and connected graphs that share a common universal cover $T$ and such that the proportion of eigenvalues of $G_n$ that lie within the support of the spectrum of $T$ tends to 1 in the large $n$ limit. This is a weak notion of being Ramanujan. We prove such a sequence of graphs is asymptotically locally tree-like. This is deduced by way of an analogous theorem proved for certain infinite sofic graphs and unimodular networks, which extends results for regular graphs and certain infinite Cayley graphs.     

Embeddings of canonical modules and resolutions of connected sums

For an ideal $I_{m,n}$ generated by all square-free monomials of degree m in a polynomial ring R with n variables, we obtain a specific embedding of a canonical module of $R/I_{m,n}$ to $R/I_{m,n}$ itself. The construction of this explicit embedding depends on a minimal free R-resolution of an ideal generated by $I_{m,n}$. Using this embedding, we give a resolution of connected sums of several copies of certain Artin $\mathsf{k}$-algebras where $\mathsf{k}$ is a field.     

More complete intersection theorems

The seminal complete intersection theorem of Ahlswede and Khachatrian gives the maximum cardinality of a $k$-uniform $t$-intersecting family on $n$ points, and describes all optimal families. In recent work, we extended this theorem to the weighted setting, giving the maximum ${\mu }_p$ measure of a $t$-intersecting family on $n$ points. In this work, we prove two new complete intersection theorems. The first gives the supremum ${\mu }_p$ measure of a $t$-intersecting family on infinitely many points, and the second gives the maximum cardinality of a subset of ${\mathbb{Z}}_m^n$ in which any two elements $x,y$ have $t$ positions $i_1,\dots ,i_t$ such that $x_{i_j}?y_{i_j}\in \{?(s?1),\dots ,s?1\}$. In both cases, we determine the extremal families, whenever possible.     

On mixing behavior of a family of random walks determined by a linear recurrence

We study random walks on the integers mod $G_n$ that are determined by an integer sequence ${\{G_n\}}_{n\ge 1}$ generated by a linear recurrence relation. Fourier analysis provides explicit formulas to compute the eigenvalues of the transition matrices and we use this to bound the mixing time of the random walks.     

The combinatorial properties of the Benoumhani polynomials for the Whitney numbers of Dowling lattices

In the papers (Benoumhani 1996;1997), Benoumhani defined two polynomials $F_{m,n,1}\left(x\right)$ and $F_{m,n,2}\left(x\right)$. Then, he defined $A_m(n,k)$ and $B_m(n,k)$ to be the polynomials satisfying $F_{m,n,1}\left(x\right)={\sum }_{k=0}^n{A}_m(n,k)x^{n?k}{(x+1)}^k$ and $F_{m,n,1}\left(x\right)={\sum }_{k=0}^n{B}_m(n,k)x^{n?k}{(x+1)}^k$. In this paper, we give a combinatorial interpretation of the coefficients of $A_{m+1}(n,k)$ and prove a symmetry of the coefficients, i.e., $\left[m^s\right]A_{m+1}(n,k)=\left[m^{n?s}\right]A_{m+1}(n,n?k)$. We give a combinatorial interpretation of $B_{m+1}(n,k)$ and prove that $B_{m+1}(n,n?1)$ is a polynomial in $m$ with non-negative integer coefficients. We also prove that if $n\ge 6$ then all coefficients of $B_{m+1}(n,n?2)$ except the coefficient of $m^{n?1}$ are non-negative integers. For all $n$, the coefficient of $m^{n?1}$ in $B_{m+1}(n,n?2)$ is $?(n?1)$, and when $n\le 5$ some other coefficients of $B_{m+1}(n,n?2)$ are also negative.     

Fractional chromatic numbers of tensor products of three graphs

The tensor product $(G_1,G_2,G_3)$ of graphs $G_1$, $G_2$ and $G_3$ is defined by $V(G_1,G_2,G_3)=V\left(G_1\right)\times V\left(G_2\right)\times V\left(G_3\right)$and $E(G_1,G_2,G_3)=\left\{\left((u_1,u_2,u_3),(v_1,v_2,v_3)\right):\left|\{i:(u_i,v_i)\in E\left(G_i\right)\}\right|\ge 2\right\}.$Let ${\chi }_f\left(G\right)$ be the fractional chromatic number of a graph $G$. In this paper, we prove that if one of the three graphs $G_1$, $G_2$ and $G_3$ is a circular clique, ${\chi }_f(G_1,G_2,G_3)=min\{{\chi }_f\left(G_1\right){\chi }_f\left(G_2\right),{\chi }_f\left(G_1\right){\chi }_f\left(G_3\right),{\chi }_f\left(G_2\right){\chi }_f\left(G_3\right)\}.$     

No-three-in-line problem on a torus: Periodicity

Let ${\tau }_{m,n}$ denote the maximal number of points on the discrete torus (discrete toric grid) of sizes $m\times n$ with no three collinear points. The value ${\tau }_{m,n}$ is known for the case where $gcd(m,n)$ is prime. It is also known that ${\tau }_{m,n}\le 2gcd(m,n)$. In this paper we generalize some of the known tools for determining ${\tau }_{m,n}$ and also show some new. Using these tools we prove that the sequence ${\left({\tau }_{z,n}\right)}_{n\in \mathbb{N}}$ is periodic for all fixed $z>1$. In general, we do not know the period; however, if $z=p^a$ for $p$ prime, then we can bound it. We prove that ${\tau }_{p^a,p^{(a?1)p+2}}=2p^a$ which implies that the period for the sequence is $p^b$, where $b$ is at most $(a?1)p+2$.