We present a new proof of the Kővári-Sós-Turán theorem that $ex(n, K_{s,t}) = O(n^{2-1/t})$ for $s, t \geq 2$. The new proof is elementary, avoiding the use of convexity. For any $d$-uniform hypergraph $H$, let $ex_d(n,H)$ be the maximum possible number of edges in an $H$-free $d$-uniform hypergraph on $n$ vertices. Let $K_{H, t}$ be the $(d+1)$-uniform hypergraph obtained from $H$ by adding $t$ new vertices $v_1, \dots, v_t$ and replacing every edge $e$ in $E(H)$ with $t$ edges $e \cup \left\{v_1\right\},\dots, e \cup \left\{v_t\right\}$ in $E(K_{H, t})$. If $H$ is the $1$-uniform hypergraph on $s$ vertices with $s$ edges, then $K_{H, t} = K_{s, t}$.
We prove that $ex_{d+1}(n,K_{H,t}) = O(ex_d(n, H)^{1/t} n^{d+1-d/t} + t n^d)$ for any $d$-uniform hypergraph $H$ with at least two edges such that $ex_d(n, H) = o(n^d)$. Thus $ex_{d+1}(n,K_{H,t}) = O(n^{d+1-1/t})$ for any $d$-uniform hypergraph $H$ with at least two edges such that $ex_d(n, H) = O(n^{d-1})$, which implies the Kővári-Sós-Turán theorem in the $d = 1$ case. This also implies that $ex_{d+1}(n, K_{H,t}) = O(n^{d+1-1/t})$ when $H$ is a $d$-uniform hypergraph with at least two edges in which all edges are pairwise disjoint, which generalizes an upper bound proved by Mubayi and Verstraëte (JCTA, 2004). We also obtain analogous bounds for 0-1 matrix Turán problems.
We start with a bijective proof of Schur's theorem due to Alladi and Gordon and describe how a particular iteration of it leads to some very general theorems on colored partitions. These theorems imply a number of important results, including Schur's theorem, Bressoud's generalization of a theorem of Göllnitz, two of Andrews' generalizations of Schur's theorem, and the Andrews-Olsson identities.
In this paper we go on to discuss about Stanley's theorem in Integer partitions. We give two different versions for the proof of the generalization of Stanley's theorem illustrating different techniques that may be applied to profitably understand the underlying structure behind the theorem.
We consider a number of generalizations of the $β$-extended MacMahon Master Theorem for a matrix. The generalizations are based on replacing permutations on multisets formed from matrix indices by partial permutations or derangements over matrix or submatrix indices.
We study a generalization of the classical Pentagonal Number Theorem and its applications. We derive new identities for certain infinite series, recurrence relations and convolution sums for certain restricted partitions and divisor sums. We also derive new identities for Bell polynomials.
In this note, we investigate some properties of local Kneser graphs defined in [8]. In this regard, as a generalization of the Erd${\rm \ddot{o}}$s-Ko-Rado theorem, we characterize the maximum independent sets of local Kneser graphs. Next, we present an upper bound for their chromatic number.
The Erdős--Ko--Rado theorem is extended to designs in semilattices with certain conditions. As an application, we show the intersection theorems for the Hamming schemes, the Johnson schemes, bilinear forms schemes, Grassmann schemes, signed sets, partial permutations and restricted signed sets.
In this paper, we describe and prove a generalization of both the classical Greene-Kleitman duality theorem for posets and the local version proved recently by Lewis-Lyu-Pylyavskyy-Sen in studying discrete solitons, using an approach more closely linked to the approach of the classical case.
We prove some analogues of the central point theorem and Tverberg's theorem, where instead of points, we consider hyperplanes or affine flats of given dimension.
The author gave a proof of a generalization of the Aztec diamond theorem for a family of $4$-vertex regions on the square lattice with southwest-to-northeast diagonals drawn in (Electron. J. Combin., 2014) by using a bijection between tilings and non-intersecting lattice paths. In this paper, we use Kuo graphical condensation to give a new proof.
In this paper, we provide a common generalization to the well-known Erdős-Ko-Rado Theorem, Frankl-Wilson Theorem, Alon-Babai-Suzuki Theorem, and Snevily Theorem on set systems with $\mathcal{L}$-intersections. As a consequence, we derive a result which strengthens substantially the well-known theorem on set systems with $k$-wise $\mathcal{L}$-intersections by F$\ddot{u}$redi and Sudakov [J. Combin. Theory, Ser. A (2004) 105: 143-159]. We will also derive similar results on $\mathcal{L}$-intersecting families of subspaces of an $n$-dimensional vector space over a finite field $\mathbb{F}_{q}$, …
In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the Zeilberger--Bressoud $q$-Dyson theorem or the $q$-Dyson constant term identity. This conjecture was proved by Károlyi, Lascoux and Warnaar in 2015. In this paper, by slightly changing the variables of Kadell's conjecture, we obtain another symmetric function generalization of the $q$-Dyson constant term identity. This new generalized constant term admits a simple product-form expression.
We generalize Aztec diamond theorem (N. Elkies, G. Kuperberg, M. Larsen, and J. Propp, Alternating-sign matrices and domino tilings, Journal Algebraic Combinatoric, 1992) by showing that the numbers of tilings of a certain family of regions in the square lattice with southwest-to-northeast diagonals drawn in are given by powers of 2. We present a proof for the generalization by using a bijection between domino tilings and non-intersecting lattice paths.
If a non-periodic sequence $X$ is the image by a morphism of a fixed point of both a primitive substitution $σ$ and a primitive substitution $τ$, then the dominant eigenvalues of the matrices of $σ$ and of $τ$ are multiplicatively dependent. This is the way we propose to generalize Cobham's Theorem.
We generalize the Yao-Yao partition theorem by showing that for any smooth measure in $R^d$ there exist equipartitions using $(t+1)2^{d-1}$ convex regions such that every hyperplane misses the interior of at least $t$ regions. In addition, we present tight bounds on the smallest number of hyperplanes whose union contains the boundary of an equipartition of a measure into $n$ regions. We also present a simple proof of a Borsuk-Ulam type theorem for Stiefel manifolds that allows us to generalize the central transversal theorem and prove results bridging the Yao--Yao partition theorem and the cen…
One of the most remarkable theorems in coding theory is Gleason's 1970 theorem about the weight enumerators of self-dual codes. In the past 36 years there have been hundreds of papers written about generalizations and applications of this theorem to different types of codes, always on a case-by-case basis. In this paper I state the theorem and then describe the far-reaching generalization that Gabriele Nebe, Eric Rains and I have developed which includes all the earlier generalizations at once. The full proof has just appeared in our book "Self-Dual Codes and Invariant Theory" (Springer, 2006)…
In this note, we prove a certain hypergraph generalization of the Balog-Szemeredi-Gowers Theorem. Our result shares some features in common with a similar such generalizsation due to Sudakov, Szemeredi and Vu, though the conclusion of our result is somewhat different.
Our main result is a new upper bound for the size of k-uniform, L-intersecting families of sets, where L contains only positive integers. We characterize extremal families in this setting. Our proof is based on the Ray-Chaudhuri--Wilson Theorem. As an application, we give a new proof for the Erdős-Ko-Rado Theorem, improve Fisher's inequality in the uniform case and give an uniform version of the Frankl-Füredi conjecture .
By studying the reciprocity property of linear Diophantine systems in light of Malcev-Neumann series, we present in this paper a new approach to and a generalization of Stanley's monster reciprocity theorem. A formula for the "error term" is given in the case when the system does not have the reciprocity property. We also give a short proof of Stanley's reciprocity theorem for linear homogeneous Diophantine systems.
In a recent paper, Dousse introduced a refinement of Siladić's theorem on partitions, where parts occur in two primary and three secondary colors. Her proof used the method of weighted words and $q$-difference equations. The purpose of this paper is to give a bijective proof of a generalization of Dousse's theorem from two primary colors to an arbitrary number of primary colors.
We prove a generalization of Frieman's $3k-3$ theorem for the sumset $$ Σ^{l}(A_1,\ldots,A_k)=\{a_{j_{1}}+\cdots+a_{j_{l}}:\,1\leq j_{1}<\cdots<j_{l}\leq k,\ a_{j_{s}}\in A_{j_{s}}\text{ for all }s\}. $$
We prove an inequality for the Kostka-Foulkes polynomials $K_{λ,μ}(q)$. As a corollary, we obtain a nontrivial lower bound for the Kostka numbers and a new proof of the Berenstein-Zelevinsky weight-multiplicity-one-criterium.
A weight-dependent generalization of the binomial theorem for noncommuting variables is presented. This result extends the well-known binomial theorem for q-commuting variables by a generic weight function depending on two integers. For a special case of the weight function, restricting it to depend on only a single integer, the noncommutative binomial theorem involves an expansion of complete symmetric functions. Another special case concerns the weight function to be a suitably chosen elliptic (i.e., doubly-periodic meromorphic) function, in which case an elliptic generalization of the binom…
We announce a new four parameter partition theorem from which the (big) theorem of Gollnitz follows by setting any one of the parameters equal to 0. This settles a problem of Andrews who asked whether there exists a result that goes beyond the partition theorem of Gollnitz. We state a four parameter q-series identity (key identity) which is the generating function form of this theorem. In a subsequent paper, the proof of the new four parameter key identity will be given.
