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A k‐uniform hypergraph with n vertices is an (n,k,ℓ)$$ \left(n,k,\ell \right) $$‐omitting system if it has no two edges with intersection size ℓ$$ \ell $$. If in addition it has no two edges with intersection size greater than ℓ$$ \ell $$, then it is an (n,k,ℓ)$$ \left(n,k,\ell \right) $$‐system. Rödl and Šiňajová proved a sharp lower bound for the independence number of (n,k,ℓ)$$ \left(n,k,\ell \right) $$‐systems. We consider the same question for (n,k,ℓ)$$ \left(n,k,\ell \right) $$‐omitting systems. Our proofs use adaptations of the random greedy independent set algorithm, and pseudorandom graphs. We also prove related results where we forbid more than two edges with a prescribed common intersection size leading to some applications in Ramsey theory. For example, we obtain good bounds for the Ramsey number rk(F,t)$$ {r}_k\left(F,t\right) $$, where F is the k‐uniform Fan. The behavior is quite different than the case k=2$$ k=2 $$ which is the classical Ramsey number r(3,t)$$ r\left(3,t\right) $$. [ABSTRACT FROM AUTHOR]

The triangle covering number of a graph is the minimum number of vertices that hit all triangles. Given positive integers s,t and an n‐vertex graph G with ⌊n2∕4⌋+t edges and triangle covering number s, we determine (for large n) sharp bounds on the minimum number of triangles in G and also describe the extremal constructions. Similar results are proved for cliques of larger size and color critical graphs. This extends classical work of Rademacher, Erdős, and Lovász–Simonovits whose results apply only to s≤t. Our results also address two conjectures of Xiao and Katona. We prove one of them and give a counterexample and prove a modified version of the other conjecture. [ABSTRACT FROM AUTHOR]

A conjecture of Chung and Graham states that every K4‐free graph on n vertices contains a vertex set of size ⌊n∕2⌋ that spans at most n2∕18 edges. We make the first step toward this conjecture by showing that it holds for all regular graphs. [ABSTRACT FROM AUTHOR]