Erdős matching conjecture for almost perfect matchings

Large Yk,b ${Y}_{k,b}$‐tilings and Hamilton ℓ $\ell $‐cycles in k $k$‐uniform hypergraphs

“…, where n k s ( , ) 0 is a constant depending on k and s. Subsequent improvements on n 0 have been done by various authors [5,20,24,35] and the current state of the art is n sk s − ≤ for sufficiently large s by Frankl and Kupavskii [24]. Kolupaev and Kupavskii [38] verified the conjecture for ( )…”

“…Subsequent improvements on $${n}_{0}$$ have been done by various authors [5, 20, 24, 35] and the current state of the art is $${n}_{0}\le \frac{5}{3}sk-\frac{2}{3}s$$ for sufficiently large $$s$$ by Frankl and Kupavskii [24]. Kolupaev and Kupavskii [38] verified the conjecture for $$n\le (s+1)(k+\frac{1}{100k})$$, improving a result of Frankl [22].…”

Generalized Turán results for disjoint cliques

“…Even the asymptotics of ex(n, K p r ) is unknown for every 2 < p < r. The exact value of ex(n, tK p p ) is known for sufficiently large n. The optimal threshold on n is stated in a conjecture of Erdős [4] and attracted a lot of researchers, see e.g. [11] and the references in it.…”