On Ramsey Theory and Slow Bootstrap Percolation

This dissertation concerns two sets of problems in extremal combinatorics. The major part, Chapters 1 to 4, is about Ramsey-type problems for cycles. The shorter second part, Chapter 5, is about a problem in bootstrap percolation. Next, we describe each topic more precisely. Given three graphs G, L1 and L2, we say that G arrows (L1, L2) and write G → (L1, L2), if for every edge-coloring of G by two colors, say 1 and 2, there exists a color i whose color class contains Li as a subgraph. The classical problem in Ramsey theory is the case where G, L1 and L2 are complete graphs; in this case the question is how large the order of G must be (in terms of the orders of L1 andL2) to guarantee that G → (L1, L2). Recently there has been much interest in the case where L1 and L2 are cycles and G is a graph whose minimum degree is large. In the past decade, numerous results have been proved about those problems. We will continue this work and prove two conjectures that have been left open. Our main weapon is Szemeredi\u27s Regularity Lemma.Our second topic is about a rather unusual aspect of the fast expanding theory of bootstrap percolation. Bootstrap percolation on a graph G with parameter r is a cellular automaton modeling the spread of an infection: starting with a set A0, cointained in V(G), of initially infected vertices, define a nested sequence of sets, A0 ⊆ A1 ⊆. . . ⊆ V(G), by the update rule that At+1, the set of vertices infected at time t + 1, is obtained from At by adding to it all vertices with at least r neighbors in At. The initial set A0 percolates if At = V(G) for some t. The minimal such t is the time it takes for A0 to percolate. We prove results about the maximum percolation time on the two-dimensional grid with parameter r = 2

Minimum lethal sets in grids and tori under 3-neighbour bootstrap percolation

Let $\mathcal{r} ≥ 1$ be any non negative integer and let $G = (V, E)$ be any undirected graph in which a subset $D ⊆ V$ of vertices are initially infected. We consider the following process in which, at every step, each non-infected vertex with at least $\mathcal{r}$ infected neighbours becomes and an infected vertex never becomes non-infected. The problem consists in determining the minimum size $s_r (G)$ of an initially infected vertices set $D$ that eventually infects the whole graph $G$. Note that $s_1 (G)$ = 1 for any connected graph $G$. This problem is closely related to cellular automata, to percolation problems and to the Game of Life studied by John Conway. Note that $s_1(G) = 1$ for any connected graph $G$. The case when $G$ is the $n × n$ grid $G_{n×n}$ and $\mathcal{r} = 2$ is well known and appears in many puzzles books, in particular due to the elegant proof that shows that $s_2(G_{n×n})$ = $n$ for all $n$ ∈ $\mathbb{N}$. We study the cases of square grids $G_{n×n}$ and tori $T_{n×n}$ when $\mathcal{r}$ ∈ {3, 4}. We show that $s_3(G_{n×n})$ = $\lceil\frac{n^2+2n+4}{3}\rceil$ for every $n$ even and that $\lceil\frac{n^2 +2n}{3}\rceil$ ≤ $s_3(G_ {n×n})$ ≤ $\lceil\frac{n^2 +2n}{3}\rceil$ + 1 for any $n$ odd. When $n$ is odd, we show that both bounds are reached, namely $s_3(G_{n×n})$ = $\lceil\frac{n^2 +2n}{3}\rceil$ if $n$ ≡ 5 (mod 6) or $n$ = 2$^p$ − 1 for any $p$ ∈ $\mathbb{N}^*$, and $s_3(G_{n×n})$ = $\lceil\frac{n^2 +2n}{3}\rceil$ + 1 if $n$ ∈ {9, 13}. Finally, for all $n$ ∈ $\mathbb{N}$, we give the exact expression of $s_4(G_{n×n})$ and of $s_r(T_{n×n})$ when $\mathcal{r}$ ∈ {3, 4}

Ramsey theory for cycles and paths

Os principais objetos de estudo neste trabalho são os números de Ramsey para circuitos e o lema da regularidade de Szemerédi. Dados grafos $L_1, \\ldots, L_k$, o número de Ramsey $R(L_1,\\ldots,L_k)$ é o menor inteiro $N$ tal que, para qualquer coloração com $k$ cores das arestas do grafo completo com $N$ vértices, existe uma cor $i$ para a qual a classe de cor correspondente contém $L_i$ como um subgrafo. Estaremos especialmente interessados no caso em que os grafos $L_i$ são circuitos. Obtemos um resultado original solucionando o caso em que $k=3$ e $L_i$ são circuitos pares de mesmo tamanho.The main objects of interest in this work are the Ramsey numbers for cycles and the Szemerédi regularity lemma. For graphs $L_1, \\ldots, L_k$, the Ramsey number $R(L_1, \\ldots,L_k)$ is the minimum integer $N$ such that for any edge-coloring of the complete graph with~$N$ vertices by $k$ colors there exists a color $i$ for which the corresponding color class contains~$L_i$ as a subgraph. We are specially interested in the case where the graphs $L_i$ are cycles. We obtained an original result solving the case where $k=3$ and $L_i$ are even cycles of the same length

The 3-colored Ramsey number of even cycles

Denote by R(L,L,L) the minimum integer N such that any 3-coloring of the edges of the complete graph on N vertices contains a monochromatic copy of a graph L. Bondy and Erdős conjectured that when L is the cycle Cn on n vertices, R(Cn,Cn,Cn)=4n−3 for every odd n>3. Łuczak proved that if n is odd, then R(Cn,Cn,Cn)=4n+o(n), as n→∞, and Kohayakawa, Simonovits and Skokan confirmed the Bondy–Erdős conjecture for all sufficiently large values of n. Figaj and Łuczak determined an asymptotic result for the ‘complementary’ case where the cycles are even: they showed that for even n, we have R(Cn,Cn,Cn)=2n+o(n), as n→∞. In this paper, we prove that there exists n1 such that for every even n⩾n1, R(Cn,Cn,Cn)=2n

Democracia na América Latina : democratização, tensões e aprendizados

É possível esboçar pistas sobre as transformações nos repertórios de ação coletiva dos movimentos sociais após o golpe de 2016. Durante o lulismo, os movimentos, por meio do “diálogo crítico”, interagiam com maior recorrência com o Estado, por meio de canais participativos. Todavia, após o golpe, abrem-se oportunidades políticas para o confronto político, possibilitando a adoção de práticas políticas disruptivas frente ao governo Temer. Não há fórmula pronta para o que está por vir no Brasil. Afinal, estamos no “tempo das incertezas”. Contudo, é possível perceber que, em que pese o país estar sofrendo uma guinada à direita, ela mesma pode abrir oportunidades para ações disruptivas de confronto, recolocando o terreno da sociedade civil como uma arena de disputas de projetos políticos e de reconfiguração das práticas políticas tradicionais