In this paper we study random linear systems with $k > 3$ variables per equation over the finite field GF(2), or equivalently k-XOR-CNF formulas. In a previous paper Creignou and Daudé proved that there exists a phase transition exhibiting a sharp threshold, for the consistency (satisfiability) of such systems (formulas). The control parameter for this transition is the ratio of the number of equations to the number of variables, and the scale for which the transition occurs remains somewhat elusive. In this paper we establish, for any $k > 3$, non-trivial lower and upper estimates of the value of the control ratio for which the phase transition occurs. For $k=3$ we get 0.89 and 0.93, respectively. Moreover, we give experimental results for $k=3$ suggesting that the critical ratio is about 0.92. Our estimates are clearly close to the critical ratio.

The results of this paper concern the ‘large spectra’ of sets, by which we mean the set of points in ${\bb F}_p^{\times}$ at which the Fourier transform of a characteristic function $\chi_A$, $A\subseteq {\bb F}_p$, can be large. We show that a recent result of Chang concerning the structure of the large spectrum is best possible. Chang's result has already found a number of applications in combinatorial number theory.

We also show that if $|A|=\lfloor {p/2}\rfloor$, and if $R$ is the set of points $r$ for which $|\hat{\chi}_A(r)|\geqslant \alpha p$, then almost nothing can be said about $R$ other than that $|R|\ll \alpha^{-2}$, a trivial consequence of Parseval's theorem.

A celebrated theorem of Turán asserts that every graph on n vertices with more than $\frac{r\,{-}\,1}{2r}n^2$ edges contains a copy of a complete graph $K_r+1$. In this paper we consider the following more general question. Let G be a $K_r+1-free graph of order n and let α be a constant, 0<α≤1. How dense can every induced subgraph of G on αn vertices be? We prove the following local density extension of Turán's theorem.

For every integer $r\geq 2$ there exists a constant $c_r < 1$ such that, if $c_r < \alpha < 1$ and every αn vertices of G span more than

\[ \frac{r\,{-}\,1}{2r}(2\alpha -1)n^2\vspace*{7pt} \]

edges, then G contains a copy of $K_r+1$. This result is clearly best possible and answers a question of Erdős, Faudree, Rousseau and Schelp [5].

In addition, we prove that the only $K_r+1-free graph of order n, in which every αn vertices span at least $\frac{r\,{-}\,1}{2r}(2\alpha -1)n^2$ edges, is a Turán graph. We also obtain the local density version of the Erdős–Stone theorem.

For an integer $s\ges 2$, a property $\P^{(s)}$ is an infinite class of s-uniform hypergraphs closed under isomorphism. We say that a property $\P^{(s)}$ is \emph{hereditary\/} if~$\P^{(s)}$ is closed under taking induced subhypergraphs. Thus, for some `forbidden class' $\FF=\{\F_i^{(s)}\:i\in I\}$ of s-uniform hypergraphs, $\P^{(s)}$ is the set of all s-uniform hypergraphs not containing any $\F_i^{(s)}\in\FF$ as an induced subhypergraph. Let $\P^{(s)}_n$ be those hypergraphs of $\P^{(s)}$ on some fixed n-vertex set. For a set of s-uniform hypergraphs $\FF=\{\F_i^{(s)}\:i\in I\}$, let

\[ \exind(n,\FF)=\max\bigl|[n]^s{\setminus}(\M\cup\N)\vphantom{\big|}\bigr|, \]

where the maximum is taken over all $\M$ and $\N\subseteq[n]^s$ with $\M\cap\N=\emptyset$ such that, for all $\G\subseteq[n]^s{\setminus}(\M\cup\N)$, no $\F_i^{(s)}\in\FF$ appears as an induced subhypergraph of $\G\cup\M$. We show that

\[ \log_2\big|\P^{(3)}_n\big|=\exind(n,\FF)+o(n^3) \]

holds for $s=3$ and any hereditary property $\P^{(3)}$, where $\FF$ is a forbidden class associated with $\P^{(3)}$. This result complements a collection of analogous theorems already proved for graphs (i.e., $s=2$).

We provide an alternative, simpler and more general derivation of the Clarkson–Shor probabilistic technique [7] and use it to obtain several extensions and new combinatorial bounds.

We investigate the asymptotic behaviour of the average displacement of the simple random walk on the Sierpiński graph. The existence of an oscillating factor in this asymptotics is shown rigorously. The proof depends mainly on the analysis of the corresponding generating function. Using a functional equation and techniques from complex analysis we obtain the desired properties of this generating function.