Related papers: On a Question of Gowers on Clique Differences

Geometric Complexity Theory as initiated by Mulmuley and Sohoni in two papers (SIAM J Comput 2001, 2008) aims to separate algebraic complexity classes via representation theoretic multiplicities in coordinate rings of specific group…

We construct classes of graphs that are variants of the so-called layered wheel. One of their key properties is that while the treewidth is bounded by a function of the clique number, the construction can be adjusted to make the dependance…

In a graph, a Clique-Stable Set separator (CS-separator) is a family $\mathcal{C}$ of cuts (bipartitions of the vertex set) such that for every clique $K$ and every stable set $S$ with $K \cap S = \emptyset$, there exists a cut $( W,W')$ in…

An abstract, Hales-Jewett type extension of the polynomial van der Waerden Theorem [J. Amer. Math. Soc. 9 (1996),725-753] is established: Theorem. Let r,d,q \in \N. There exists N \in \N such that for any r-coloring of the set of subsets of…

We give a block decomposition of the dg category of character sheaves on a simple and simply-connected complex reductive group $G$, similar to the one in generalized Springer correspondence. As a corollary, we identify the category of…

The generic homomorphism problem, which asks whether an input graph $G$ admits a homomorphism into a fixed target graph $H$, has been widely studied in the literature. In this article, we provide a fine-grained complexity classification of…

Hadwiger's conjecture asserts that any graph contains a clique minor with order no less than the chromatic number of the graph. We prove that this well-known conjecture is true for all graphs if and only if it is true for squares of split…

One of the central problems in additive combinatorics is to determine how large a subset of the first $N$ integers can be before it is forced to contain $k$ elements forming an arithmetic progression. Around 25 years ago, Gowers proved the…

The bandwidth of a graph G is the minimum of the maximum difference between adjacent labels when the vertices have distinct integer labels. We provide a polynomial algorithm to produce an optimal bandwidth labeling for graphs in a special…

Clique-width is a well-studied graph parameter. For graphs of bounded clique-width, many problems that are NP-hard in general can be polynomial-time solvable. The fact motivates several studies to investigate whether the clique-width of…

We show that various identities from [1] and [3] involving Gould-Hopper polynomials can be deduced from the real but also complex orthogonal invariance of multivariate Gaussian distributions. We also deduce from this principle a useful…

The Cluster Deletion problem takes a graph $G$ as input and asks for a minimum size set of edges $X$ such that $G-X$ is the disjoint union of complete graphs. An equivalent formulation is the Clique Partition problem, which asks to find a…

We prove that Sherali-Adams with polynomially bounded coefficients requires proofs of size $n^{\Omega(d)}$ to rule out the existence of an $n^{\Theta(1)}$-clique in Erd\H{o}s-R\'{e}nyi random graphs whose maximum clique is of size $d\leq…

There has been recent interest in lower bounds for the principal specializations of Schubert polynomials $\nu_w := \mathfrak S_w(1,\dots,1)$. We prove a conjecture of Yibo Gao in the setting of $1243$-avoiding permutations that gives a…

We study the density of the weights of Generalized Reed--Muller codes. Let $RM_p(r,m)$ denote the code of multivariate polynomials over $\F_p$ in $m$ variables of total degree at most $r$. We consider the case of fixed degree $r$, when we…

In this paper, we introduce (almost) skew 2-nomial algebras and look for a one-sided or two-sided Gr\"obner basis theory for such algebras at a modest level. That is, we establish the existence of a skew multiplicative $K$-basis for every…

I present a single algorithm which solves the clique problems, "What is the largest size clique?", "What are all the maximal cliques?" and the decision problem, "Does a clique of size k exist?" for any given graph in polynomial time. The…

The class P is in fact a proper sub-class of NP. We explore topological properties of the Hamming space 2^[n] where [n]={1, 2,..., n}. With the developed theory, we show: (i) a theorem that is closely related to Erdos and Rado's sunflower…

Given a graph G, let Q(G) denote the collection of all independent (edge-free) sets of vertices in G. We consider the problem of determining the size of a largest antichain in Q(G). When G is the edge-less graph, this problem is resolved by…

Sperner's bound on the size of an antichain in the lattice P(S) of subsets of a finite set S has been generalized in three different directions: by Erdos to subsets of P(S) in which chains contain at most r elements; by Meshalkin to certain…