On the graph limit question of Vera T. S\'os

In the dense graph limit theory, the topology of the set of graphs is defined by the distribution of the subgraphs spanned by finite number of random vertices. Vera T. S\'os proposed a question that if we consider only the number of edges in the spanned subgraphs, then whether it provides an equivalent definition. We show that the answer is positive on quasirandom graphs, and we prove a generalization of the statement.Comment: 4 page

General removal lemma

We formulate and prove a general result in spirit of hypergraph removal lemma for measurable functions of several variables.Comment: 5 page

Pentagons in triangle-free graphs

For all $n\ge 9$, we show that the only triangle-free graphs on $n$ vertices maximizing the number $5$-cycles are balanced blow-ups of a 5-cycle. This completely resolves a conjecture by Erd\H{o}s, and extends results by Grzesik and Hatami, Hladk\'y, Kr\'{a}l', Norin and Razborov, where they independently showed this same result for large $n$ and for all $n$ divisible by $5$.Comment: 6 page

On the Typical Structure of Graphs in a Monotone Property

Given a graph property $\mathcal{P}$, it is interesting to determine the typical structure of graphs that satisfy $\mathcal{P}$. In this paper, we consider monotone properties, that is, properties that are closed under taking subgraphs. Using results from the theory of graph limits, we show that if $\mathcal{P}$ is a monotone property and $r$ is the largest integer for which every $r$-colorable graph satisfies $\mathcal{P}$, then almost every graph with $\mathcal{P}$ is close to being a balanced $r$-partite graph.Comment: 5 page

Remarks on Graphons

The notion of the graphon (a symmetric measurable fuzzy set of $[0, 1]^2$) was introduced by L. Lov\'asz and B. Szegedy in 2006 to describe limit objects of convergent sequences of dense graphs. In their investigation the integral $$t(F,W)=\int _{[0, 1]^k}\prod _{ij\in E(F)}W(x_i,x_j)dx_1dx_2\cdots dx_k$$ plays an important role in which $W$ is a graphon and $E(F)$ denotes the set of all edges of a $k$-labelled simple graph $F$. In our present paper we show that the set of all fuzzy sets of $[0, 1]^2$ is a right regular band with respect to the operation $\circ$ defined by $$(f\circ g)(s,t)=\vee _{(x,y)\in [0, 1]^2}(f(x,y)\wedge g(s,t));\quad (s, t)\in [0, 1]^2,$$ and the set of all graphons is a left ideal of this band. We prove that, if $W$ is an arbitrary graphon and $f$ is a fuzzy set of $[0, 1]^2$, then $$|t(F; W)-t(F; f\circ W)|\leq |E(F)|(\sup(W)-\sup(f))\Delta (\{W> \sup(f)\} )$$ for arbitrary finite simple graphs $F$, where $\Delta (\{W> \sup(f)\})$ denotes the area of the set $\{W>\sup(f)\}$ of all $(x, y)\in [0, 1]^2$ satisfying $W(x,y)>\sup(f)$.Comment: 11 page

The Hoffmann-Jorgensen inequality in metric semigroups

We prove a refinement of the inequality by Hoffmann-Jorgensen that is significant for three reasons. First, our result improves on the state-of-the-art even for real-valued random variables. Second, the result unifies several versions in the Banach space literature, including those by Johnson and Schechtman [Ann. Probab. 17 (1989)], Klass and Nowicki [Ann. Probab. 28 (2000)], and Hitczenko and Montgomery-Smith [Ann. Probab. 29 (2001)]. Finally, we show that the Hoffmann-Jorgensen inequality (including our generalized version) holds not only in Banach spaces but more generally, in a very primitive mathematical framework required to state the inequality: a metric semigroup $\mathscr{G}$. This includes normed linear spaces as well as all compact, discrete, or (connected) abelian Lie groups.Comment: 11 pages, published in the Annals of Probability. The Introduction section shares motivating examples with arXiv:1506.0260