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Abstract Bollobás and Riordan, in their paper ‘Metrics for sparse graphs’, proposed a number of provocative conjectures extending central results of quasirandom graphs and graph limits to sparse graphs. We refute these conjectures by exhibiting a sequence of graphs with convergent normalized subgraph densities (and pseudorandom C 4 -counts), but with no limit expressible as a kernel. 

Abstract Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon–Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor’s result to nonabelian groups. 

Abstract May the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ 4– o (1) but not O ( δ 4 ). Let M ( δ ) be the maximum number such that the following holds: for every ∊ > 0 and $G = {\mathbb{F}}_2^n$ with n sufficiently large, if A ⊆ G × G with A ≥ δ | G | 2 , then there exists a nonzero “popular difference” d ∈ G such that the number of “corners” ( x , y ), ( x + d , y ), ( x , y + d ) ∈ A is at least ( M ( δ )–∊)| G | 2 . As a corollary via a recent result of Mandache, we conclude that M ( δ ) = δ 4– o (1) and M ( δ ) = ω ( δ 4 ). On the other hand, for 0 < δ < 1/2 and sufficiently large N , there exists A ⊆ [ N ] 3 with | A | ≥ δN 3 such that for every d ≠ 0, the number of corners ( x , y , z ), ( x + d , y , z ), ( x , y + d , z ), ( x , y , z + d ) ∈ A is at most δ c log(1/ δ ) N 3 . A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3.