In this paper we present a simple unifying approach to prove several statements about intersecting and cross-intersecting families, including the Erdős-Ko-Rado theorem, the Hilton-Milner theorem, a theorem due to Frankl concerning the size of intersecting families with bounded maximal degree, and versions of results on the sum of sizes of non-empty cross-intersecting families due to Frankl and Tokushige. Several new stronger results are also obtained.Our approach is based on the use of regular bipartite graphs. These graphs are quite often used in Extremal Set Theory problems, however, the approach we develop proves to be particularly fruitful.
In this short note, we prove the following analog of the Kővári–Sós–Turán theorem for intersection graphs of boxes. If G is the intersection graph of n axis-parallel boxes in
$${{\mathbb{R}}^d}$$
such that G contains no copy of K
t,t
, then G has at most ctn( log n)2d+3 edges, where c = c(d)>0 only depends on d. Our proof is based on exploring connections between boxicity, separation dimension and poset dimension. Using this approach, we also show that a construction of Basit, Chernikov, Starchenko, Tao and Tran of K2,2-free incidence graphs of points and rectangles in the plane can be used to disprove a conjecture of Alon, Basavaraju, Chandran, Mathew and Rajendraprasad. We show that there exist graphs of separation dimension 4 having superlinear number of edges.
The Danzer-Grünbaum acute angles problem asks for the largest size of a set of points in R d that determines only acute angles. Recently, the problem was essentially solved thanks to the results of the second author and of Gerencsér and Harangi: now, the lower and the upper bounds are 2 d−1 + 1 and 2 d − 1, respectively. The lower-bound construction is surprisingly simple.In this note, we suggest the following variant of the problem, which is one way to "save" the problem. Put F (α) = lim d→∞ f (d, α) 1/d , where f (d, α) is the largest set of points in R d with no angle greater than α. Then the question is to find c := lim α→π/2 − F (α). Although one may expect that c = 2 in view of the result of Gerencsér and Harangi, the best lower bound we could get is c √ 2. We also solve a related problem of Erdős and Füredi on the "stability" of the acute angles problem and refute another conjecture stated in the same paper.
We construct skew corner-free sets in [n] 2 of size n 5/4 , thereby disproving a conjecture of Kevin Pratt. We also show that any skew corner-free set in F n q × F n q must have size at most q (2−c)n , for some positive constant c which depends on q.
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