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A hypergraph $mathcal{F}$ is non-trivial intersecting if every two edges in it have a nonempty intersection but no vertex is contained in all edges of $mathcal{F}$. Mubayi and Verstra{e}te showed that for every $k ge d+1 ge 3$ and $n ge (d+1)n/d$ every $k$-graph $mathcal{H}$ on $n$ vertices without a non-trivial intersecting subgraph of size $d+1$ contains at most $binom{n-1}{k-1}$ edges. They conjectured that the same conclusion holds for all $d ge k ge 4$ and sufficiently large $n$. We confirm their conjecture by proving a stronger statement. They also conjectured that for $m ge 4$ and sufficiently large $n$ the maximum size of a $3$-graph on $n$ vertices without a non-trivial intersecting subgraph of size $3m+1$ is achieved by certain Steiner systems. We give a construction with more edges showing that their conjecture is not true in general.
This exposition contains a short and streamlined proof of the recent result of Kwan, Letzter, Sudakov and Tran that every triangle-free graph with minimum degree $d$ contains an induced bipartite subgraph with average degree $Omega(ln d/lnln d)$.
Motzkin and Straus established a remarkable connection between the maximum clique and the Lagrangian of a graph in 1965. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number
For positive integers $n,r,k$ with $nge r$ and $kge2$, a set ${(x_1,y_1),(x_2,y_2),dots,(x_r,y_r)}$ is called a $k$-signed $r$-set on $[n]$ if $x_1,dots,x_r$ are distinct elements of $[n]$ and $y_1dots,y_rin[k]$. We say a $t$-intersecting family cons
A family of subsets of $[n]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran independe
Let $kge 3$ be an odd integer and let $n$ be a sufficiently large integer. We prove that the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $2$-regular subgraphs is $binom{n-1}{k-1} + lfloorfrac{n-1}{k} rfloor$, and the