# Combinatorics Seminar

When: Sunday, March 26, 10am

Where: Schreiber 309

Speaker: Mykhaylo Tyomkyn (Tel Aviv University)

Title: Lagrangians of hypergraphs: The Frankl-Furedi conjecture holds almost everywhere

## Abstract:

Frankl and Furedi conjectured in 1989 that the maximum Lagrangian of all r-uniform hypergraphs of given size m is realised by the initial segment of the colexicographic order. For r=3 this was partially solved by Talbot, but for r\geq 4 the conjecture was widely open. We verify the conjecture for all r\geq 4, whenever

$\binom{t-1}{r} \leq m \leq \binom{t}{r}- \gamma_r t^{r-2}$

for a constant $\gamma_r>0$. This range includes the principal case $m=\binom{t}{r}$ for large enough $t$

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