We investigate Boolean degree 1 functions for several classical association schemes, including Johnson graphs, Grassmann graphs, graphs from polar spaces, and bilinear forms graphs, as well as some other domains such as multislices (Young subgroups of the symmetric group). In some settings, Boolean degree 1 functions are also known as completely regular strength 0 codes of covering radius 1, Cameron-Liebler line classes, and tight sets.We classify all Boolean degree 1 functions on the multislice. On the Grassmann scheme Jq(n, k) we show that all Boolean degree 1 functions are trivial for n ≥ 5, k, n − k ≥ 2 and q ∈ {2, 3, 4, 5}, and that, for general q, the problem can be reduced to classifying all Boolean degree 1 functions on Jq(n, 2). We also consider polar spaces and the bilinear forms graphs, giving evidence that all Boolean degree 1 functions are trivial for appropriate choices of the parameters.1 (also strength 0 designs) [44]. Motivated by problems on permutation groups and finite geometry, Boolean degree 1 functions are also known as tight sets [2,12] and Cameron-Liebler line classes [17]. The history of Cameron-Liebler line classes is particularly complicated, as the problem was introduced by Cameron and Liebler [7], the term coined by Penttila [52,53], and the algebraic point of view as Boolean degree 1 functions only emerged later; see [59], in particular §3.3.1, for a discussion of this.Due to this variation of terminology, classification results of Boolean degree 1 functions on J(n, k) were obtained repeatedly in the literature (with small variations due to different definitions) at least three times, in [49] for completely regular strength 0 codes of covering radius 1, in [25] for Boolean degree 1 functions, and in [11] for Cameron-Liebler line classes:Theorem 1.2 (Folklore). Suppose that k, n − k ≥ 2. Every Boolean degree 1 function on J(n, k) is either constant or depends on a single coordinate.Depending on the definition used, this is either easy to observe or requires a more elaborate proof. For the hypercube H(n, 2), which is a product domain, and the Johnson graph J(n, k) classifying Boolean degree 1 functions is trivial, but there are various other classical association schemes for which classification is more difficult. The Grassmann scheme J q (n, k) consists of all k-spaces of F n q as vertices, two vertices being adjacent if their meet is a subspace of dimension k − 1. Boolean degree 1 functions on J q (4, 2) were intensively investigated, and many non-trivial examples [6,8,9,24,33,37] and existence conditions [34,47] are known.We call 1-dimensional subspaces of F n q points, 2-dimensional subspaces of F n q lines, and (n − 1)dimensional subspaces of F n q hyperplanes. For a point p we define p + (S) = 1 p∈S and p − (S) = 1 p / ∈S , and for a hyperplane π we define π + (S) = 1 S⊆π and π − (S) = 1 S π . The following was shown by Drudge for q = 3 [17, Theorem 6.4]; by Gavrilyuk and Mogilnykh for q = 4 [35, Theorem 3]; and by Gavrilyuk and Matkin [32,45] for q = 5; the result for q = 2 fol...

In this paper a new parameter for hypergraphs called hypergraph infection is defined. This concept generalizes zero forcing in graphs to hypergraphs. The exact value of the infection number of complete and complete bipartite hypergraphs is determined. A formula for the infection number for interval hypergraphs and several families of cyclic hypergraphs is given. The value of the infection number for a hypergraph whose edges form a symmetric t-design is given, and bounds are determined for a hypergraph whose edges are a t-design. Finally, the infection number for several hypergraph products and line graphs are considered.

We show that a Boolean degree d function on the slice [n] k = {(x 1 , . . . , x n ) ∈ {0, 1} : n i=1 x i = k} is a junta, assuming that k, n − k are large enough. This generalizes a classical result of Nisan and Szegedy on the hypercube. Moreover, we show that the maximum number of coordinates that a Boolean degree d function can depend on is the same on the slice and the hypercube.

We prove a conjecture by Van Dam & Sotirov on the smallest eigenvalue of (distance-j) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-j) Johnson graphs. More generally, we study the smallest eigenvalue and the second largest eigenvalue in absolute value of the graphs of the relations of classical P -and Q-polynomial association schemes.1 are adjacent if they differ in exactly j elements. As for the Hamming scheme, these graphs provide examples for which the performance ratio of the Goemans-Williamson algorithm is tight and their smallest eigenvalues are central for determining their max-cuts [20]. These graphs are also important for investigating subsets with exactly one forbidden intersection, a variation of the classical Erdős-Ko-Rado theorem due to Frankl and Füredi [18].The other graphs under investigation are Grassmann graphs, dual polar graphs, and various forms graphs, most prominently the bilinear forms graphs. Again, the smallest eigenvalues can be used to investigate the max-cuts and intersecting families in these graphs. The P -polynomial graphs obtain their importance from various applications. For example, Grassmann graphs are of interest due to their applications in network coding theory [26] and their role in the recent proof of the 2-to-2-games conjecture [21].In the following we give a short summary of our main results on the specific families.

A construction of Alon and Krivelevich gives highly pseudorandom K k -free graphs on n vertices with edge density equal to Θ(n −1/(k−2) ). In this short note we improve their result by constructing an infinite family of highly pseudorandom K k -free graphs with a higher edge density of Θ(n

We describe a general construction of strongly regular graphs from the collinearity graph of a finite classical polar spaces of rank at least 3 over a finite field of order q. We show that these graphs are non-isomorphic to the collinearity graphs and have the same parameters. To our knowledge for most of these parameters these graphs are new as the collinearity graphs were the only known examples.

Two results are obtained that give upper bounds on partial spreads and partial ovoids respectively.The first result is that the size of a partial spread of the Hermitian polar space H(3, q 2 ) is at most 2p 3 +p 3 t +1, where q = p t , p is a prime. For fixed p this bound is in o(q 3 ), which is asymptotically better than the previous best known bound of (q 3 + q + 2)/2. Similar bounds for partial spreads of H(2d − 1, q 2 ), d even, are given. The second result is that the size of a partial ovoid of the Ree-Tits octagon O(2 t ) is at most 26 t + 1. This bound, in particular, shows that the Ree-Tits octagon O(2 t ) does not have an ovoid.

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