Hypergraph Lagrangians I: the Frankl-Füredi conjecture is false

Lagrangian densities of short 3-uniform linear paths and Turán numbers of their extensions

“…Subsequent progress in this conjecture were made in the papers of Tang, Peng, Zhang and Zhao [21], Tyomkyn [22], and Lei, Lu and Peng [12]. Recently, Gruslys, Letzter and Morrison [6] confirmed this conjecture for r = 3 and the number of edges is sufficiently large.…”

Lagrangians of Hypergraphs II: When colex is best

“…The most noteworthy such work, apart from the above progress on the r = 3 case, is a paper by Tyomkyn [24] who proved the conjecture for r ≥ 4 and 'almost all' values of m. Other relevant works on the case r ≥ 4 include Nikiforov [17], Lu [15] and Lei and Lu [13]. See [10] for a more detailed overview of their work.…”

“…The proof of Theorem 1.3 relies on exploiting the relationship between the problem of maximising the Lagrangian and the problem of maximising the sum of degrees squared. We remark that this relationship is also explored in [10], where we used it to find an infinite family of counterexamples to (1.2) for r ≥ 4.…”

“…In Section 2.1 we provide notation and definitions, in Section 2.2 we state some relevant results from previous papers, and in Section 2.3 we provide additional preliminary results, along with their proofs. We note that all of the content of Sections 2.1 and 2.2 appears in [10] together with the proofs. We say that w is a weighting of…”

“…The result of Motzkin and Straus [16] confirms the truth of this conjecture when r = 2. In [10], we disproved this conjecture by finding an infinite family of counterexamples for all r ≥ 4. In this paper we resolve the remaining case, that is, when r = 3.…”