On a Theorem of Sewell and Trotter

Sewell and Trotter [J. Combin. Theory Ser. B, 1993] proved that every connected alpha-critical graph that is not isomorphic to K_1, K_2 or an odd cycle contains a totally odd K_4-subdivision. Their theorem implies an interesting min-max relation for stable sets in graphs without totally odd K_4-subdivisions. In this note, we give a simpler proof of Sewell and Trotter's theorem.Comment: Referee comments incorporate

The critical window for the classical Ramsey-Tur\'an problem

The first application of Szemer\'edi's powerful regularity method was the following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.Comment: 34 page

Problems in extremal graph theory

We consider a variety of problems in extremal graph and set theory. The {\em chromatic number} of $G$, $\chi(G)$, is the smallest integer $k$ such that $G$ is $k$-colorable. The {\it square} of $G$, written $G^2$, is the supergraph of $G$ in which also vertices within distance 2 of each other in $G$ are adjacent. A graph $H$ is a {\it minor} of $G$ if $H$ can be obtained from a subgraph of $G$ by contracting edges. We show that the upper bound for $\chi(G^2)$ conjectured by Wegner (1977) for planar graphs holds when $G$ is a $K_4$-minor-free graph. We also show that $\chi(G^2)$ is equal to the bound only when $G^2$ contains a complete graph of that order. One of the central problems of extremal hypergraph theory is finding the maximum number of edges in a hypergraph that does not contain a specific forbidden structure. We consider as a forbidden structure a fixed number of members that have empty common intersection as well as small union. We obtain a sharp upper bound on the size of uniform hypergraphs that do not contain this structure, when the number of vertices is sufficiently large. Our result is strong enough to imply the same sharp upper bound for several other interesting forbidden structures such as the so-called strong simplices and clusters. The {\em $n$-dimensional hypercube}, $Q_n$, is the graph whose vertex set is $\{0,1\}^n$ and whose edge set consists of the vertex pairs differing in exactly one coordinate. The generalized Tur\'an problem asks for the maximum number of edges in a subgraph of a graph $G$ that does not contain a forbidden subgraph $H$. We consider the Tur\'an problem where $G$ is $Q_n$ and $H$ is a cycle of length $4k+2$ with $k\geq 3$. Confirming a conjecture of Erd{\H o}s (1984), we show that the ratio of the size of such a subgraph of $Q_n$ over the number of edges of $Q_n$ is $o(1)$, i.e. in the limit this ratio approaches 0 as $n$ approaches infinity

Two conjectures in Ramsey-Tur\'an theory

Given graphs $H_1,\ldots, H_k$, a graph $G$ is $(H_1,\ldots, H_k)$-free if there is a $k$-edge-colouring $\phi:E(G)\rightarrow [k]$ with no monochromatic copy of $H_i$ with edges of colour $i$ for each $i\in[k]$. Fix a function $f(n)$, the Ramsey-Tur\'an function $\textrm{RT}(n,H_1,\ldots,H_k,f(n))$ is the maximum number of edges in an $n$-vertex $(H_1,\ldots,H_k)$-free graph with independence number at most $f(n)$. We determine $\textrm{RT}(n,K_3,K_s,\delta n)$ for $s\in\{3,4,5\}$ and sufficiently small $\delta$, confirming a conjecture of Erd\H{o}s and S\'os from 1979. It is known that $\textrm{RT}(n,K_8,f(n))$ has a phase transition at $f(n)=\Theta(\sqrt{n\log n})$. However, the values of $\textrm{RT}(n,K_8, o(\sqrt{n\log n}))$ was not known. We determined this value by proving $\textrm{RT}(n,K_8,o(\sqrt{n\log n}))=\frac{n^2}{4}+o(n^2)$, answering a question of Balogh, Hu and Simonovits. The proofs utilise, among others, dependent random choice and results from graph packings.Comment: 20 pages, 2 figures, 2 pages appendi

String Spectrum of 1+1-Dimensional Large N QCD with Adjoint Matter

We propose gauging matrix models of string theory to eliminate unwanted non-singlet states. To this end we perform a discretised light-cone quantisation of large N gauge theory in 1+1 dimensions, with scalar or fermionic matter fields transforming in the adjoint representation of SU(N). The entire spectrum consists of bosonic and fermionic closed-string excitations, which are free as N tends to infinity. We analyze the general features of such bound states as a function of the cut-off and the gauge coupling, obtaining good convergence for the case of adjoint fermions. We discuss possible extensions of the model and the search for new non-critical string theories.Comment: 20 pages (7 figures available from authors as postscipt files), PUPT-134

Structure Factors and Their Distributions in Driven Two-Species Models

We study spatial correlations and structure factors in a three-state stochastic lattice gas, consisting of holes and two oppositely ``charged'' species of particles, subject to an ``electric'' field at zero total charge. The dynamics consists of two nearest-neighbor exchange processes, occuring on different times scales, namely, particle-hole and particle-particle exchanges. Using both, Langevin equations and Monte Carlo simulations, we study the steady-state structure factors and correlation functions in the disordered phase, where density profiles are homogeneous. In contrast to equilibrium systems, the average structure factors here show a discontinuity singularity at the origin. The associated spatial correlation functions exhibit intricate crossovers between exponential decays and power laws of different kinds. The full probability distributions of the structure factors are universal asymmetric exponential distributions.Comment: RevTex, 18 pages, 4 postscript figures included, mistaken half-empty page correcte

Supersaturation Problem for Color-Critical Graphs

The \emph{Tur\'an function} \ex(n,F) of a graph $F$ is the maximum number of edges in an $F$-free graph with $n$ vertices. The classical results of Tur\'an and Rademacher from 1941 led to the study of supersaturated graphs where the key question is to determine $h_F(n,q)$, the minimum number of copies of $F$ that a graph with $n$ vertices and \ex(n,F)+q edges can have. We determine $h_F(n,q)$ asymptotically when $F$ is \emph{color-critical} (that is, $F$ contains an edge whose deletion reduces its chromatic number) and $q=o(n^2)$. Determining the exact value of $h_F(n,q)$ seems rather difficult. For example, let $c_1$ be the limit superior of $q/n$ for which the extremal structures are obtained by adding some $q$ edges to a maximum $F$-free graph. The problem of determining $c_1$ for cliques was a well-known question of Erd\H os that was solved only decades later by Lov\'asz and Simonovits. Here we prove that $c_1>0$ for every {color-critical}~$F$. Our approach also allows us to determine $c_1$ for a number of graphs, including odd cycles, cliques with one edge removed, and complete bipartite graphs plus an edge.Comment: 27 pages, 2 figure