Abstract Let left bracket q right bracket equals StartSet 0 comma 1 comma ellipsis comma q minus 1 EndSet [ q ] = { 0 , 1 , … , q − 1 } $[q] = \{0,1,\ldots ,q-1\}$ , let normal upper Delta left bracket q right bracket Δ [ q ] $\Delta [q]$ denote the simplex of probability measures on left bracket q right bracket [ q ] $[q]$ , and let gamma γ $\gamma$ denote the Lebesgue measure normalized on normal upper Delta left bracket q right bracket Δ [ q ] $\Delta [q]$ . We prove that for any symmetric monotone function f colon left bracket q right bracket Superscript n Baseline right arrow left bracket q right bracket f : [ q ] n → [ q ] ${\kern1pt}f \colon{\kern-1pt} [q]^n \to [q]$ and any a element of left bracket q right bracket a ∈ [ q ] $a \in [q]$ , we have γ ( { μ ∈ Δ [ q ] | P x ∼ μ ⊗ n [ f ( x ) = a ] ∈ ( ε , 1 − ε ) } ) = O ( 1 / log n ) . \begin{equation*} \gamma (\{\mu \in \Delta [q]\;\vert \;\mathbb{P}_{x\sim \mu ^{\otimes n}}[\,f(x)=a] \in (\varepsilon ,1-\varepsilon )\}) = O(1/\log n)\text{.} \end{equation*} We also show that this bound is tight. This improves Kalai and Mossel's previous bound of upper O left parenthesis log log n slash log n right parenthesis O ( log log n / log n ) $O(\!\log \log n/\log n)$ and answers their question completely.
Let Omega(1), ... , Omega(m) be probability spaces, let Omega =Omega(1) & times; & centerdot;& centerdot;& centerdot; & times; Omega(m) be their product and let A(1), ... , A(n) subset of Omega be events. Suppose that each event A(i) depends on ri coordinates of a point x is an element of Q, x = (xi(1), ... ,xi(m)), and that for each event A(i) there are A(i) other events A(j) that depend on some of the coordinates that A(i) depends on. Let A = max{5, Ai: i= 1, ... , n} and let & micro;(i) = min{r(i), Delta(i) + 1} for i = 1, ... , n. We prove that if P(A(i)) < (3 Delta)(-3 & micro;i) for all i, then for any 0 < E < 1, the probability P boolean AND(n)(i=1) (A) over bar (-)(i) of the intersection of the complements of all Ai can be computed within relative error is an element of in polynomial time from the probabilities P (A(i1) boolean AND ... boolean AND A(ik)) of k-wise intersections of the events A(i) for k = e(O(Delta)) ln (n/is an element of).
Let $r, k, n$ be integers satisfying $1\leqslant r\leqslant k\leqslant n/2$ . Let ${{\mathcal{R}}}_r(n, k)$ denote the proportion of permutations $\pi \in {{\mathcal{S}}}_n$ that fix a set of size $k$ and have no cycle of length less than $r$ . In this note, we determine the order of magnitude of ${{\mathcal{R}}}_r(n, k)$ uniformly for all $2\leqslant r\leqslant k\leqslant n/2$ . This result generalises the corresponding estimate of Eberhard, Ford, and Green for the case $r=1$ .
We prove that several natural graph classes have tree-decompositions with minimum width such that each bag has bounded treewidth. For example, every planar graph has a tree-decomposition with minimum width such that each bag has treewidth at most 3. This treewidth bound is best possible. More generally, every graph of Euler genus g has a tree-decomposition with minimum width such that each bag has treewidth in O(g). This treewidth bound is best possible. Most generally, every K_p-minor-free graph has a tree-decomposition with minimum width such that each bag has treewidth at most some polynomial function f(p). In such results, the assumption of an excluded minor is justified, since we show that analogous results do not hold for the class of 1-planar graphs, which is one of the simplest non-minor-closed monotone classes. In fact, we show that 1-planar graphs do not have tree-decompositions with width within an additive constant of optimal, and with bags of bounded treewidth. On the other hand, we show that 1-planar n-vertex graphs have tree-decompositions with width O(√(n)) (which is the asymptotically tight bound) and with bounded treewidth bags. Moreover, this result holds in the more general setting of bounded layered treewidth, where the union of a bounded number of bags has bounded treewidth.
We show that there exist constants delta(1), delta(2) > 0 such that if G is an (n, d, lambda)-graph with lambda/d <= delta(1), then G contains an induced cycle of length at least delta 2n/d. We further demonstrate that, up to a constant factor, this is best possible. Utilising our techniques, we derive that the number of non-isomorphic induced subgraphs of such G is at least exponential in n log d/d, and further demonstrate that this is tight up to a constant factor in the exponent.
