Related papers: Sunflowers in set systems of bounded dimension
We prove an analog of the flower theorem for non-degenerate reduced tangent to the identity germs that fix the coordinate hyperspaces in any dimension.
A family of sets is intersecting if every pair of its sets intersect. A star is a family with some element (a center) in each of its sets. The classical 1961 result of Erd\H{o}s, Ko, and Rado states that every intersecting family of r-sets…
A family of sets is union-closed (UC) if the union of any two sets in the family is also in the family. Frankl's UC sets conjecture states that for any nonempty UC family $\mathcal{F} \subseteq 2^{[n]}$ such that $\mathcal{F} \neq…
Let ${\cal T}=(T,w)$ be a weighted finite tree with leaves $1,..., n$.For any $I :=\{i_1,..., i_k \} \subset \{1,...,n\}$,let $D_I ({\cal T})$ be the weight of the minimal subtree of $T$ connecting $i_1,..., i_k$; the $D_{I} ({\cal T})$ are…
Let F be a set system on [n] with all sets having k elements and every pair of sets intersecting. The celebrated theorem of Erdos-Ko-Rado from 1961 says that any such system has size at most ${n-1 \choose k-1}$. A natural question, which…
Given a family $\mathcal{F}=\{A_1,\dots,A_s\}$ of subsets of $\mathbb{Z}_n$, define $\Delta \mathcal{F}$ to be the multiset of all (cyclic) distances dist$(x,y)$, where $\{x,y\} \subset A_i$, $x \neq y$, for some $i=1,\dots,s$. Taking…
We show three basic properties on the image Milnor number $\mu_I(f)$ of a germ $f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with isolated instability. First, we show the conservation of the image Milnor number, from which one…
We prove the following the generalized Tur\'an type result. A collection $\mathcal{T}$ of $r$ sets is an $r$-triangle if for every $T_1,T_2,\dots,T_{r-1}\in \mathcal{T}$ we have $\cap_{i=1}^{r-1}T_i\neq\emptyset$, but $\cap_{T\in…
A $(v,k,\lambda)$-BIBD $(X,\mathcal B)$ can be nested if there is a mapping $\phi:\mathcal B\rightarrow X$ such that $(X,\{B\cup\{\phi(B)\}\mid B\in\mathcal B\})$ is a $(v,k+1,\lambda+1)$-packing. A $(v,k,\lambda)$-BIBD has a (perfect)…
We say that a family of $k$-subsets of an $n$-element set is intersecting if any two of its sets intersect. In this paper we study properties and structure of large intersecting families. We prove a conclusive version of Frankl's theorem on…
We study a finite-field analogue of the Erd\H{o}s distinct distances problem under the Hamming metric. For a set \(S\subseteq \mathbb{F}_q^n\) let $\Delta(S)$ denote the set of Hamming distances determined by \(S\). We prove the lower bound…
Let $k>1$, and let $\mathcal{F}$ be a family of $2n+k-3$ non-empty sets of edges in a bipartite graph. If the union of every $k$ members of $\mathcal{F}$ contains a matching of size $n$, then there exists an $\mathcal{F}$-rainbow matching…
In this paper we show that every set $A \subset \mathbb{N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb{N}$, settling a conjecture of Erd\H{o}s. The proof features two different decompositions of…
The Katz-Sarnak density conjecture states that, as the analytic conductor $R \to \infty$, the distribution of the normalized low-lying zeros (those near the central point $s = 1/2$) converges to the scaling limits of eigenvalues clustered…
The Gyarfas-Sumner conjecture asserts that if H is a tree then every graph with bounded clique number and very large chromatic number contains H as an induced subgraph. This is still open, although it has been proved for a few simple…
We prove that any family $E_1, \ldots , E_{\lceil rn \rceil}$ of (not necessarily distinct) sets of edges in an $r$-uniform hypergraph, each having a fractional matching of size $n$, has a rainbow fractional matching of size $n$ (that is, a…
A $k$-cycle with a pendant edge attached to each vertex is called a $k$-sun. The existence problem for $k$-sun decompositions of $K_v$, with $k$ odd, has been solved only when $k=3$ or $5$. By adapting a method used by Hoffmann, Lindner and…
In 1965, Paul Erd\H{o}s asked about the largest family $Y$ of $k$-sets in $\{ 1, \ldots, n \}$ such that $Y$ does not contain $s+1$ pairwise disjoint sets. This problem is commonly known as the Erd\H{o}s Matching Conjecture. We investigate…
We show a general result known as the Erdos_Sos Conjecture: if $E(G)>{1/2}(k-1)n$ where $G$ has order $n$ then $G$ contains every tree of order $k+1$ as a subgraph.
A rough structure theorem is proved for graphs $G$ containing no copy of a bounded degree tree $T$: from any such $G$, one can delete $o(|G||T|)$ edges in order to get a subgraph all of whose connected components have a cover of order…