Subsets of Products of Finite Sets of Positive Upper Density

In this note we prove that for every sequence $(m_q)_{q}$ of positive integers and for every real $0<\delta\leqslant1$ there is a sequence $(n_q)_{q}$ of positive integers such that for every sequence $(H_q)_{q}$ of finite sets such that $|H_q|=n_q$ for every $q\in\mathbb{N}$ and for every $D\subseteq \bigcup_k\prod_{q=0}^{k-1}H_q$ with the property that $$\limsup_k \frac{|D\cap \prod_{q=0}^{k-1} H_q|}{|\prod_{q=0}^{k-1}H_q|}\geqslant\delta$$ there is a sequence $(J_q)_{q}$, where $J_q\subseteq H_q$ and $|J_q|=m_q$ for all $q$, such that $\prod_{q=0}^{k-1}J_q\subseteq D$ for infinitely many $k.$ This gives us a density version of a well-known Ramsey-theoretic result. We also give some estimates on the sequence $(n_q)_{q}$ in terms of the sequence of $(m_q)_{q}$.Comment: 12 page

Strong Forms of Stability from Flag Algebra Calculations

Given a hereditary family $\mathcal{G}$ of admissible graphs and a function $\lambda(G)$ that linearly depends on the statistics of order-$\kappa$ subgraphs in a graph $G$, we consider the extremal problem of determining $\lambda(n,\mathcal{G})$, the maximum of $\lambda(G)$ over all admissible graphs $G$ of order $n$. We call the problem perfectly $B$-stable for a graph $B$ if there is a constant $C$ such that every admissible graph $G$ of order $n\ge C$ can be made into a blow-up of $B$ by changing at most $C(\lambda(n,\mathcal{G})-\lambda(G)){n\choose2}$ adjacencies. As special cases, this property describes all almost extremal graphs of order $n$ within $o(n^2)$ edges and shows that every extremal graph of order $n\ge n_0$ is a blow-up of $B$. We develop general methods for establishing stability-type results from flag algebra computations and apply them to concrete examples. In fact, one of our sufficient conditions for perfect stability is stated in a way that allows automatic verification by a computer. This gives a unifying way to obtain computer-assisted proofs of many new results.Comment: 44 pages; incorporates reviewers' suggestion

A density version of the Carlson--Simpson theorem

We prove a density version of the Carlson--Simpson Theorem. Specifically we show the following. For every integer $k\geq 2$ and every set $A$ of words over $k$ satisfying $$\limsup_{n\to\infty} \frac{|A\cap [k]^n|}{k^n}>0$$ there exist a word $c$ over $k$ and a sequence $(w_n)$ of left variable words over $k$ such that the set $$\{c\}\cup \big\{c^{\smallfrown}w_0(a_0)^{\smallfrown}...^{\smallfrown}w_n(a_n) : n\in\mathbb{N} \ \text{ and } \ a_0,...,a_n\in [k]\big\}$$ is contained in $A$. While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.Comment: 73 pages, no figure