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One of the most intruguing conjectures in extremal graph theory is the conjecture of Erdős and Sós from 1962, which asserts that every $$n$$-vertex graph with more than $$\frac{k-1}{2}n$$ edges contains any $$k$$-edge tree as a subgraph. Kalai proposed a generalization of this conjecture to hypergraphs. To explain the generalization, we need to define the concept of a tight tree in an $$r$$-uniform hypergraph, i.e., a hypergraph where each edge contains $$r$$ vertices. A tight tree is an $$r$$-uniform hypergraph such that there is an ordering $$v_1,\ldots,v_n$$ of its its vertices with the following property: the vertices $$v_1,\ldots,v_r$$ form an edge and for every $i>r$, there is a single edge $$e$$ containing the vertex $$v_i$$ and $r-1$ of the vertices $$v_1,\ldots,v_{i-1}$$, and $$e\setminus\{v_i\}$$ is a subset of one of the edges consisting only of vertices from $$v_1,\ldots,v_{i-1}$$. The conjecture of Kalai asserts that every $$n$$-vertex $$r$$-uniform hypergraph with more than $$\frac{k-1}{r}\binom{n}{r-1}$$ edges contains every $$k$$-edge tight tree as a subhypergraph. The recent breakthrough results on the existence of combinatorial designs by Keevash and by Glock, Kühn, Lo and Osthus show that this conjecture, if true, would be tight for infinitely many values of $$n$$ for every $$r$$ and $$k$$.The article deals with the special case of the conjecture when the sought tight tree is a path, i.e., the edges are the $$r$$-tuples of consecutive vertices in the above ordering. The case $r=2$ is the famous Erdős-Gallai theorem on the existence of paths in graphs. The case $r=3$ and $k=4$ follows from an earlier work of the authors on the conjecture of Kalai. The main result of the article is the first non-trivial upper bound valid for all $$r$$ and $$k$$. The proof is based on techniques developed for a closely related problem where a hypergraph comes with a geometric structure: the vertices are points in the plane in a strictly convex position and the sought path has to zigzag beetwen the vertices.
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A completion of the proof of the Edge-statistics Conjecture
Extremal combinatorics often deals with problems of maximizing a specific quantity related to substructures in large discrete structures. The first question of this kind that comes to one's mind is perhaps determining the maximum possible number of induced subgraphs isomorphic to a fixed graph $$H$$ in an $$n$$-vertex graph. The asymptotic behavior of this number is captured by the limit of the ratio of the maximum number of induced subgraphs isomorphic to $$H$$ and the number of all subgraphs with the same number vertices as $$H$$; this quantity is known as the _inducibility_ of $$H$$. More generally, one can define the inducibility of a family of graphs in the analogous way.Among all graphs with $$k$$ vertices, the only two graphs with inducibility equal to one are the empty graph and the complete graph. However, how large can the inducibility of other graphs with $$k$$ vertices be? Fix $$k$$, consider a graph with $$n$$ vertices join each pair of vertices independently by an edge with probability $$\binom{k}{2}^{-1}$$. The expected number of $$k$$-vertex induced subgraphs with exactly one edge is $$e^{-1}+o(1)$$. So, the inducibility of large graphs with a single edge is at least $$e^{-1}+o(1)$$. This article establishes that this bound is the best possible in the following stronger form, which proves a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn: the inducibility of the family of $$k$$-vertex graphs with exactly $$l$$ edges where $$0<\binom{k}{2}$$ is at most $$e^{-1}+o(1)$$. The example above shows that this is tight for $l=1$ and it can be also shown to be tight for $l=k-1$. The conjecture was known to be true in the regime where $$l$$ is superlinearly bounded away from $$0$$ and $$\binom{k}{2}$$, for which the sum of the inducibilities goes to zero, and also in the regime where $$l$$ is bounded away from $$0$$ and $$\binom{k}{2}$$ by a sufficiently large linear function. The article resolves the hardest cases where $$l$$ is linearly close to $$0$$ or close to $$\binom{k}{2}$$, and provides generalizations to hypergraphs.
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- Award ID(s):
- 1855635
- PAR ID:
- 10178072
- Date Published:
- Journal Name:
- Advances in Combinatorics
- Volume:
- 1
- Issue:
- 1
- ISSN:
- 2517-5599
- Page Range / eLocation ID:
- 52 pages
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.19086/aic.12047
