Hypergraph Lagrangians I: the Frankl-F\"uredi conjecture is false

An old and well-known conjecture of Frankl and F\"{u}redi states that the Lagrangian of an $r$-uniform hypergraph with $m$ edges is maximised by an initial segment of colex. In this paper we disprove this conjecture by finding an infinite family of counterexamples for all $r \ge 4$. We also show that, for sufficiently large $t \in \mathbb{N}$, the conjecture is true in the range $\binom{t}{r} \le m \le \binom{t+1}{r} - \binom{t-1}{r-2}$.Comment: We split our original paper (arXiv:1807.00793v2) into two parts. This first part consists of 24 pages, including a one-page appendix. The second part appears in a new submission (arXiv:1907.09797

Perturbation theory of transformed quantum fields

We consider a scalar quantum field $\phi$ with arbitrary polynomial self-interaction in perturbation theory. If the field variable $\phi$ is repaced by a local diffeomorphism $\phi(x) = \rho(x) + a_1 \rho^2(x) +\ldots$, this field $\rho$ obtains infinitely many additional interaction vertices. We show that the $S$-matrix of $\rho$ coincides with the one of $\phi$ without using path-integral arguments. This result holds even if the underlying field has a propagator of higher than quadratic order in the momentum. If tadpole diagrams vanish, the diffeomorphism can be tuned to cancel all contributions of an underlying $\phi^s$-type self interaction at one fixed external offshell momentum, rendering $\rho$ a free theory at this momentum. Finally, we propose one way to extend the diffeomorphism to a non-local transformation involving derivatives without spoiling the combinatoric structure of the local diffeomorphism.Comment: 28 pages, 9 figure

Strong Jumps and Lagrangians of Non-Uniform Hypergraphs

The hypergraph jump problem and the study of Lagrangians of uniform hypergraphs are two classical areas of study in the extremal graph theory. In this paper, we refine the concept of jumps to strong jumps and consider the analogous problems over non-uniform hypergraphs. Strong jumps have rich topological and algebraic structures. The non-strong-jump values are precisely the densities of the hereditary properties, which include the Tur\'an densities of families of hypergraphs as special cases. Our method uses a generalized Lagrangian for non-uniform hypergraphs. We also classify all strong jump values for $\{1,2\}$-hypergraphs.Comment: 19 page