Strong stability of 3-wise tt-intersecting families

Let $\mathcal G$ be a family of subsets of an $n$-element set. The family $\mathcal G$ is called $3$-wise $t$-intersecting if the intersection of any three subsets in $\mathcal G$ is of size at least $t$. For a real number $p\in(0,1)$ we define the measure of the family by the sum of $p^{|G|}(1-p)^{n-|G|}$ over all $G\in\mathcal G$. For example, if $\mathcal G$ consists of all subsets containing a fixed $t$-element set, then it is a $3$-wise $t$-intersecting family with the measure $p^t$. For a given $\delta>0$, by choosing $t$ sufficiently large, the following holds for all $p$ with $0<p\leq 2/(\sqrt{4t+9}-1)$. If $\mathcal G$ is a $3$-wise $t$-intersecting family with the measure at least $(\frac12+\delta)p^t$, then $\mathcal G$ satisfies one of (i) and (ii): (i) every subset in $\mathcal G$ contains a fixed $t$-element set, (ii) every subset in $\mathcal G$ contains at least $t+2$ elements from a fixed $(t+3)$-element set

Non-trivial 3-wise intersecting uniform families

A family of $k$-element subsets of an $n$-element set is called 3-wise intersecting if any three members in the family have non-empty intersection. We determine the maximum size of such families exactly or asymptotically. One of our results shows that for every $\epsilon>0$ there exists $n_0$ such that if $n>n_0$ and $\frac25+\epsilon<\frac kn<\frac 12-\epsilon$ then the maximum size is $4\binom{n-4}{k-3}+\binom{n-4}{k-4}$.Comment: 12 page

Binding numbers and f-factors of graphs

AbstractLet G be a connected graph of order n, a and b be integers such that 1 ≤ a ≤ b and 2 ≤ b, and f: V(G) → {a, a + 1, …, b} be a function such that Σ(f(x); x ∈ V(G)) ≡ 0 (mod 2). We prove the following two results: (i) If the binding number of G is greater than (a + b −1)(n−1)(an−(a + b) + 3) and n ≥(a + b)2a, then G has an f-factor; (ii) If the minimum degree of G is greater than (bn − 2)(a + b), and n ≥(a + b)2a, then G has an f-factor